Shapley–Folkman lemma#Formal statement

A set is said to be convex if every line segment joining two of its points is a subset in the set. The solid disk $\backslash bullet$ is a convex set, but the circle $\backslash circ$ is not, because the line segment joining two distinct points $\backslash oslash$ is not a subset of the circle.{{sfnp|Takayama|1985|p=[https://books.google.com/books?id=j6PLOBFotPQC&pg=PA20 20]}} The convex hull of a set $Q$ is the smallest convex set that contains $Q$.{{sfnp|Rockafellar|1997|p=12}}

Minkowski addition is an operation on sets that forms the set of sums of members of the sets, with one member from each set.{{sfnp|Schneider|1993|loc=Chapter 3: Minkowski addition|pp=126–196}} For example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two:$$\backslash \{0,\; 1\backslash \}\; +\; \backslash \{0,\; 1\backslash \}\; =\; \backslash \{0\; +\; 0,\; 0\; +\; 1,\; 1\; +\; 0,\; 1\; +\; 1\backslash \}\; =\; \backslash \{0,\; 1,\; 2\backslash \}.$$ This subset of the integers $\backslash \{0,1,2\backslash \}$ is contained in the interval of real numbers $[0,2]$, which is its convex hull. The Shapley–Folkman lemma implies that every point in $[0,2]$ is the sum of an integer from $\backslash \{0,1\backslash \}$ and a real number from $[0,1]$: to get the convex hull of the Minkowski sum of $\backslash \{0,1\backslash \}$ with itself, only one of the summands needs to be replaced by its convex hull.{{sfnp|Carter|2001|p=94}}

The non-convex set $\backslash \{0,1\backslash \}$ and its convex hull $[0,1]$ are at Hausdorff distance $\backslash tfrac12$ from each other,

and this distance remains the same for the non-convex set $\backslash \{0,1,2\backslash \}$ and its convex hull $[0,2]$. In both cases the convex hull contains points such as $\backslash tfrac12$ that are at distance $\backslash tfrac12$ from the members of the non-convex set. In this example, the Minkowski sum operation does not decrease the distance between the sum and its convex hull. But when summing is replaced by averaging, by scaling the sum by the number of terms in the sum, the distance between the scaled Minkowski sum and its convex hull does go down. The distance between the average Minkowski sum

$$\backslash frac\{1\}\{2\}\; \backslash left(\; \backslash \{0,1\backslash \}\; +\; \backslash \{0,1\backslash \}\; \backslash right)\; =\; \backslash left\backslash \{\; 0,\; \backslash tfrac\{1\}\{2\},\; 1\; \backslash right\backslash \}$$

and its convex hull $[0,1]$ is only $\backslash tfrac14$, which is half the distance $\backslash tfrac12$ between its summand $\backslash \{0,1\backslash \}$ and its convex hull $[0,1]$. As more sets are added together, the average of their sum "fills out" its convex hull: The maximum distance between the average and its convex hull approaches zero as the average includes more summands. This reduction in distance can be stated more formally as the Shapley–Folkman theorem and Shapley–Folkman–Starr theorem, as consequences of the Shapley–Folkman lemma.{{sfnp|Carter|2001|p=94}}

The Shapley–Folkman lemma depends upon the following definitions and results from convex geometry.

A real vector space of two dimensions can be given a Cartesian coordinate system in which every point is identified by an ordered pair of real numbers, called "coordinates", which are conventionally denoted by $x$ and $y$. Two points in the Cartesian plane can be added coordinate-wise:

$$(x\_1,y\_1)\; +\; (x\_2,y\_2)\; =\; (x\_1\; +\; x\_2,\; y\_1\; +\; y\_2);$$

further, a point can be multiplied by each real number $\backslash lambda$ coordinate-wise:

$$\backslash lambda\; (x,\; y)\; =\; (\backslash lambda\; x,\; \backslash lambda\; y).$$

More generally, any real vector space of (finite) dimension $D$ can be viewed as the set of all $D$-tuples of $D$ real numbers $(x\_1,x\_2,\backslash ldots,x\_D)$ on which two operations are defined: vector addition and multiplication by a real number. For finite-dimensional vector spaces, the operations of vector addition and real-number multiplication can each be defined coordinate-wise, following the example of the Cartesian plane.{{harvtxt|Arrow|Hahn|1980|p=375}}

{{multiple image

| width = 155

| footer = Line segments test whether a subset is convex.

| image1 = Convex polygon illustration1.svg

| alt1 = Illustration of a convex set, which looks somewhat like a disk: A (green) convex set contains the (black) line-segment joining the points x and y. The entire line-segment is a subset of the convex set.

| caption1 = In a convex set {{mvar|Q}}, the line segment connecting any two of its points is a subset of {{mvar|Q}}.

| image2 = Convex polygon illustration2.svg

| alt2 = Illustration of a green non-convex set, which looks somewhat like a boomerang or cashew nut. The black line-segment joins the points x and y of the green non-convex set. Part of the line segment is not contained in the green non-convex set.

| caption2 = In a non-convex set {{mvar|Q}}, a point in some line-segment joining two of its points is not a member of {{mvar|Q}}.

}}

In a real vector space, a non-empty set $Q$ is defined to be convex if, for each pair of its points, every point on the line segment that joins them is still in $Q$. For example, a solid disk $\backslash bullet$ is convex but a circle $\backslash circ$ is not, because it does not contain a line segment joining opposite points.{{sfnp|Takayama|1985|p=[https://books.google.com/books?id=j6PLOBFotPQC&pg=PA20 20]}} A solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non-convex. The empty set is convex, either by definition{{sfnp|Rockafellar|1997|p=10}} or vacuously.{{sfnp|Blumson|2019}}

More formally, a set $Q$ is convex if, for all points $q\_1$ and $q\_2$ in $Q$ and for every real number $\backslash lambda$ in the unit interval $[0,1]$, the point

$$(1\; -\; \backslash lambda)\; q\_1\; +\; \backslash lambda\; q\_2$$

is a member of $Q$. By mathematical induction, a set $Q$ is convex if and only if every convex combination of members of $Q$ also belongs to $Q$. By definition, a convex combination of indexed points $v\_1,v\_2,\backslash ldots\; v\_D$ of a vector space is any weighted average $\backslash lambda\_1\; v\_1+\backslash lambda\_2\; v\_2+\backslash cdots+\backslash lambda\_D\; v\_D$ for indexed real numbers $\backslash lambda\_1,\backslash lambda\_2,\backslash ldots\; \backslash lambda\_D$ satisfying the equation $\backslash lambda\_1+\backslash lambda\_2+\backslash cdots+\backslash lambda\_D=1$.{{harvtxt|Arrow|Hahn|1980|p=376}}, {{harvtxt|Rockafellar|1997|pp=10–11}}, and {{harvtxt|Green|Heller|1981|p=37}}

The definition of a convex set implies that the intersection of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set. In particular, the intersection of two disjoint sets is the empty set, which is convex.{{sfnp|Rockafellar|1997|p=10}}

File:Extreme points.svg of the red set, each blue point is a convex combination of some red points.]]

For every subset $Q$ of a real vector space, its convex hull $\backslash operatorname\{conv\}Q$ is the minimal convex set that contains $Q$. Thus $\backslash operatorname\{conv\}\; Q$ is the intersection of all the convex sets that cover $Q$.{{sfnp|Rockafellar|1997|p=12}} The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in $Q$.{{harvtxt|Arrow|Hahn|1980|p=385}} and {{harvtxt|Rockafellar|1997|pp=11–12}} For example, the convex hull of the set of integers $\backslash \{0,1\backslash \}$ is the closed interval of real numbers $[0,1]$, which has the maximum and minimum of the given set as its endpoints.{{sfnp|Carter|2001|p=94}} The convex hull of the unit circle is the closed unit disk, which contains the unit circle and its interior.{{sfnp|Boules|2021|p=[https://books.google.com/books?id=Gx0gEAAAQBAJ&pg=PA83 83]}}

File:Minkowski sum graph - vector version.svg of sets. The sum of the squares $Q\_\{1\}=[0,1]^\{2\}$ and $Q\_\{2\}=[1,2]^\{2\}$ is the square $Q\_\{1\}+Q\_\{2\}=[1,3]^\{2\}$.]]

In any vector space (or algebraic structure with addition), $X$, the Minkowski sum of two non-empty sets $A,B\backslash subseteq\; X$ is defined to be the element-wise operation{{harvtxt|Schneider|1993|p=xi}} and {{harvtxt|Rockafellar|1997|p=16}} $$A+B=\backslash \{x+y\backslash mid\; x\backslash in\; A,y\backslash in\; B\backslash \}.$$ For example,

\begin{align}

\{0,1\}+\{0,1\}&=\{0+0,0+1,1+0,1+1\}\\

&=\{0,1,2\}.

\end{align}

This operation is clearly commutative and associative on the collection of non-empty sets. All such operations extend in a well-defined manner to recursive forms $\backslash sum\_\{n=1\}^\{N\}Q\_\{n\}=Q\_\{1\}+Q\_\{2\}+\backslash ldots+Q\_\{N\}.$ By the principle of induction,{{harvtxt|Rockafellar|1997|p=17}} and {{harvtxt|Starr|1997|p=78}}

$$\backslash sum\_\{n=1\}^\{N\}Q\_\{n\}=\backslash left\backslash \{\backslash sum\_\{n=1\}^\{N\}q\_\{n\}\backslash mathrel\{\backslash Bigg|\}q\_\{n\}\backslash in\; Q\_\{n\},~1\backslash leq\; n\backslash leq\; N\backslash right\backslash \}.$$

Minkowski addition behaves well with respect to taking convex hulls. Specifically, for all subsets $A,B\backslash subseteq\; X$ of a real vector space, $X$, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls. That is,

$$\backslash operatorname\{conv\}\; (A+B)\; =\; (\backslash operatorname\{conv\}A)+(\backslash operatorname\{conv\}B).$$

And by induction it follows that

$$\backslash operatorname\{conv\}\backslash sum\_\{n=1\}^\{N\}Q\_\{n\}\; =\; \backslash sum\_\{n=1\}^\{N\}\backslash operatorname\{conv\}Q\_\{n\}$$

for any $N\backslash in\backslash N$ and non-empty subsets $Q\_n\backslash in\; X,\backslash \; 1\backslash le\; n\backslash le\; N$.{{harvtxt|Schneider|1993|pp=2–3}}{{harvtxt|Arrow|Hahn|1980|p=387}}

In the following statements, $D$ and $D$ represent positive integers. $D$ is the dimension of the ambient space $\backslash R^D$.

$Q\_1,\backslash dots,Q\_N$ represent nonempty, bounded subsets of $\backslash R^D$. They are also called "summands". $N$ is the number of summands.

$Q\; =\; \backslash sum\_\{n=1\}^N\; Q\_n$ denotes the Minkowski sum of the summands.

The variable $x$ is used to represent an arbitrary vector in $\backslash operatorname\{conv\}Q$.

Because the convex hull and Minkowski sum operations can always be interchanged, as $\backslash operatorname\{conv\}Q\; =\; \backslash sum\_\{n=1\}^\{N\}\backslash operatorname\{conv\}(Q\_\{n\})$, it follows that for every $x\backslash in\backslash operatorname\{conv\}Q$, there exist elements $q\_n\backslash in\backslash operatorname\{conv\}Q\_n$ such that $\backslash sum\_\{n=1\}^\{N\}q\_\{n\}=x$. The Shapley–Folkman lemma refines this statement.{{sfnp|Carter|2001|p=93}}

{{Math theorem

| name = Shapley–Folkman lemma

| note =

| math_statement = For every $x\backslash in\backslash operatorname\{conv\}Q$, there exist elements $q\_n$ such that $\backslash sum\_\{n=1\}^\{N\}q\_\{n\}=x$, and such that at most $D$ of these elements $q\_n$ belong to $(\backslash operatorname\{conv\}Q\_n)\backslash setminus\; Q\_n$, while each remaining element $q\_n$ belongs to $Q\_n$.{{sfnp|Carter|2001|p=93}}

}}

For example, every point in $[0,2]=[0,1]+[0,1]=\backslash operatorname\{conv\}\backslash \{0,1\backslash \}+\backslash operatorname\{conv\}\backslash \{0,1\backslash \}$ is the sum of an element in $\backslash \{0,1\backslash \}$ and an element in $[0,1]$.{{sfnp|Carter|2001|p=94}}

Shuffling indices if necessary, this means that every point in $\backslash operatorname\{conv\}Q$ can be decomposed as

$$x\; =\; \backslash sum\_\{n=1\}^\{D\}q\_\{n\}\; +\; \backslash sum\_\{n=D+1\}^\{N\}\; q\_n$$

where $q\_n\backslash in\backslash operatorname\{conv\}Q\_n$ for $1\backslash le\; n\backslash le\; D$ and $q\_n\backslash in\backslash operatorname\; Q\_n$ for $D+1\backslash le\; n\backslash le\; N$. Note that the reindexing depends on the point $x$.{{harvtxt|Starr|1969|pp=35–36}}

The lemma may be stated succinctly as

$$\backslash operatorname\{conv\}\backslash left(\backslash sum\_\{n=1\}^\{N\}Q\_\{n\}\backslash right)\backslash subseteq\backslash bigcup\_\{I\backslash subseteq\backslash \{1,2,\backslash ldots\; N\backslash \}:~|I|=D\}$$

\left(

\sum_{n\in I}\operatorname{conv}Q_{n}+\sum_{n\notin I}Q_{n}\right).

{{Math theorem

| name = converse of Shapley–Folkman lemma

| note =

| math_statement = If a vector space obeys the Shapley–Folkman lemma for a natural number $D$, and for no number less than $D$, then its dimension is finite, and exactly $D$.{{harvtxt|Schneider|1993|p=140}} credits this result to {{harvtxt|Borwein|O'Brien|1978}}

}}

In particular, the Shapley–Folkman lemma requires the vector space to be finite-dimensional.

Shapley and Folkman used their lemma to prove the following theorem, which quantifies the difference between $Q$ and $\backslash operatorname\{conv\}Q$ using Hausdorff distance. Hausdorff distances measure how close two sets are. For two sets $X$ and $Y$, the Hausdorff distance is, intuitively, the smallest amount by which each must be expanded to cover the other. More formally,

if the distance from a point $x$ to a set $Y$ is defined as the infimum of pairwise distances,

$$d(x,\; Y)\; =\; \backslash inf\_\{y\backslash in\; Y\}\backslash |x-y\backslash |,$$

then let $X\_\{\backslash varepsilon\}$ denote the set of all points within

distance $\backslash varepsilon$ of $X$; equivalently this is closure of the Minkowski sum of $X$ with a ball of radius $\backslash varepsilon$. Then for $X\backslash subset\; Y$, the Hausdorff distance is{{harvtxt|Papadopoulos|2005|loc=4.1 The Hausdorff distance|pp=[https://books.google.com/books?id=JrwzXZB0YrIC&pg=PA105 105–110]}}. The formula for Hausdorff distance is simplified here using the assumption that $X\backslash subset\; Y$ to avoid requiring also that $X\backslash subset\; Y\_\{\backslash varepsilon\}$.

$$d\_\{\backslash mathrm\{H\}\}(X,Y)=\backslash inf\; \backslash \{\backslash varepsilon\backslash mid\; X\_\{\backslash varepsilon\}\backslash supset\; Y\backslash \}.$$

The Shapley–Folkman theorem quantifies how close to convexity $Q$ is by upper-bounding its Hausdorff distance to $\backslash operatorname\{conv\}Q$, using the circumradii of its constituent sets. For any bounded set $S\; \backslash subset\; \backslash R^D,$ define its circumradius $\backslash operatorname\{rad\}\; S$ to be the smallest radius of a ball containing it. More formally, letting $x$ denote the center of the smallest enclosing ball, it can be defined as{{sfnp|Balestro|Martini|Teixeira|2024|p=[https://books.google.com/books?id=xwYUEQAAQBAJ&pg=PA151 151]}}

$$\backslash operatorname\{rad\}\; S=\; \backslash inf\backslash \{\backslash varepsilon\backslash mid\backslash exists\; x\; \backslash in\; \backslash mathbb\{R\}^N:\; \backslash \{x\backslash \}\_\{\backslash varepsilon\}\backslash supset\; S\backslash \}.$$ With this notation in place, the Shapley–Folkman theorem can be stated as:

{{Math theorem

| name = Shapley–Folkman theorem

| note =

| math_statement = $d\_\{\backslash mathrm\{H\}\}(\; Q,\; \backslash operatorname\{conv\}Q)^2\; ~\backslash leq~\; \backslash sum\_\{\backslash max\; D\}\; (\backslash operatorname\{rad\}Q\_n)^2.${{harvtxt|Starr|1969|p=36}}{{harvtxt|Schneider|1993|p=129}}

}}

Here the notation $\backslash sum\_\{\backslash max\; D\}$ means "the sum of the $D$ largest terms". This upper bound depends on the dimension of ambient space and the shapes of the summands, but not on the number of summands.{{sfnp|Starr|1981}}

File:Inner radius.svg

The Shapley–Folkman theorem can be strengthened by replacing the circumradius of the terms in a Minkowski by a smaller value, the inner radius, which intuitively measures the radius of the holes in a set rather than the radius of a set.

Define the inner radius $r(S)$ of a bounded subset $S\; \backslash subset\; \backslash R^D$ to be the infimum of $r$ such that, for any $x\backslash in\; \backslash operatorname\{conv\}S$, there exists a ball $B$ of radius $r$ such that $x\backslash in\; \backslash operatorname\{conv\}(S\; \backslash cap\; B)$.{{harvtxt|Starr|1969|p=37}}

{{Math theorem

| name = Shapley–Folkman–Starr theorem

| note =

| math_statement = $d\_\{\backslash mathrm\{H\}\}(Q,\; \backslash operatorname\{conv\}Q)^2\; \backslash leq\; \backslash sum\_\{\backslash max\; D\}\; r(Q\_n)^2$.{{harvtxt|Schneider|1993|pp=129–130}}

}}

There have been many proofs of these results, from the original, to the later Arrow and Hahn,{{harvtxt|Arrow|Hahn|1980|pp=392–395}} Cassels,{{harvtxt|Cassels|1975|pp=435–436}} Schneider,{{harvtxt|Schneider|1993|p=128}} etc. An abstract and elegant proof by Ekeland{{harvtxt|Ekeland|1999|pp=357–359}} has been extended by Artstein.{{harvtxt|Artstein|1980|p=180}} Different proofs have also appeared in unpublished papers.{{sfnp|Anderson|2005}} An elementary proof of the Shapley–Folkman lemma can be found in the book by Bertsekas, together with applications in estimating the duality gap in separable optimization problems and zero-sum games.{{sfnp|Bertsekas|2009}}

Usual proofs of these results are nonconstructive: they establish only the existence of the representation, but do not provide an algorithm for computing the representation. In 1981, Starr published an iterative algorithm for a less sharp version of the Shapley–Folkman–Starr theorem.{{sfnp|Starr|1981}}

The following proof of Shapley–Folkman lemma is from {{harvtxt|Zhou|1993}}. The proof idea is to lift the representation of $x$ from $\backslash R^\{D\}$to $\backslash R^\{D+N\}$, use Carathéodory's theorem for conic hulls, then drop back to $\backslash R^D$.{{sfnp|Zhou|1993}}

{{Math proof|title=Proof of Shapley–Folkman lemma|proof=

For each $n$, represent $q\_n\; \backslash in\; \backslash operatorname\{conv\}Q\_n$ as $q\_n\; =\; \backslash sum\_\{k=1\}^K\; w\_\{n,\; k\}\; q\_\{n,k\}$, where $K$ is a large finite number, $w\_\{n,\; k\}\; \backslash geq\; 0$, and $\backslash sum\_k\; w\_\{n,\; k\}\; =\; 1$.

Now "lift" the representation $x\; =\; \backslash sum\_n\; \backslash sum\_k\; w\_\{n,k\}q\_\{n,k\}$ from $\backslash R^D$ to $\backslash R^\{D+N\}$. Define

$$\backslash bar\; x\; =\; (x,\; 1,\; ...,\; 1);\backslash quad\; \backslash bar\; q\_\{n,k\}\; =\; (q\_\{n,k\},\; e\_n)$$

where $e\_n$ is the vector in $\backslash R^N$ that has 1 at coordinate $n$, and 0 at all other coordinates.

With this, we have a lifted representation

$$\backslash bar\; x\; =\; \backslash sum\_\{n\}\backslash sum\_k\; w\_\{n,k\}\; \backslash bar\; q\_\{n,k\}.$$

That is, $\backslash bar\; x$ is in the conic hull of $\backslash \{\backslash bar\; q\_\{n,\; k\}\backslash \}\_\{n\backslash in\; 1:N,\; k\backslash in\; 1:K\}$.

By Carathéodory's theorem for conic hulls, we have an alternative representation

$$\backslash bar\; x\; =\; \backslash sum\_\{n\}\backslash sum\_k\; w\text{'}\_\{n,k\}\backslash bar\; q\_\{n,k\}$$

such that $w\text{'}\_\{n,k\}\; \backslash geq\; 0$, and at most $N+D$ of them are nonzero. Since we defined

$$\backslash bar\; x\; =\; (x,\; 1,\; ...,\; 1);\backslash quad\; \backslash bar\; q\_\{n,k\}\; =\; (q\_\{n,k\},\; e\_n)$$

this alternative representation is also a representation for $x$.

We argue that for any $n\_0\; \backslash in\; 1:N$, there must be at least one value of $k$ for which $w\text{'}\_\{n\_0,\; k\}$ is nonzero. Remember that we defined $(\backslash bar\; x)\_\{D\; +\; n\_0\}$, the $(D\; +\; n\_0)$ entry of $\backslash bar\; x$, to be $1$. At the same time, from the lifted representation of $\backslash bar\; x$,

$$(\backslash bar\; x)\_\{D\; +\; n\_0\}\; =\; \backslash left(\backslash sum\_\{n\}\; \backslash sum\_\{k\}\; w\text{'}\_\{n,k\}\; \backslash bar\; q\_\{n,k\}\backslash right)\_\{D+n\_0\}$$

$$=\; \backslash sum\_\{n\}\; \backslash sum\_\{k\}\; w\text{'}\_\{n,k\}\; (\backslash bar\; q\_\{n,k\})\_\{D+n\_0\}.$$

We drop all terms on the r.h.s. for which $n\; \backslash neq\; n\_0$ since they are zero. The remaining terms take the form $w\text{'}\_\{n\_0,k\}\; (\backslash bar\; q\_\{n\_0,k\})\_\{D+n\_0\}\; =\; w\text{'}\_\{n\_0,\; k\}$, so we find the equation

$$1\; =\; \backslash sum\_k\; w\text{'}\_\{n\_0,\; k\}.$$

It follows that there is at least one element of the sum on the r.h.s. that is non-zero.

Combining the fact that for each value of $n$ there is a non-zero $w\_\{n,k\}\text{'}$, together with the fact that there are at most $N+D$ of $w\_\{n,k\}\text{'}$ that are nonzero, we conclude that there can only be at most $D$ elements of $n\backslash in\; 1:N$ for which there are at least two of $w\_\{n,k\}\text{'}$ that are nonzero.

Thus we obtain a representation

$$\backslash bar\; x\; =\; \backslash sum\_\{n\}\; \backslash left(\backslash sum\_k\; w\text{'}\_\{n,k\}\; \backslash bar\; q\_\{n,k\}\backslash right)$$

where for at most $D$ of $n$, the term $\backslash sum\_k\; w\text{'}\_\{n,k\}\; \backslash bar\; q\_\{n,k\}$ is not in $Q\_n$.

}}

The following "probabilistic" proof of Shapley–Folkman–Starr theorem is from {{harvtxt|Cassels|1975}}.

We can interpret $\backslash operatorname\{conv\}S$ in probabilistic terms: $\backslash forall\; x\; \backslash in\; \backslash operatorname\{conv\}S$, since $x\; =\; \backslash sum\; w\_n\; q\_n$ for some $q\_n\; \backslash in\; S$, we can define a random vector $X$, finitely supported in $S$, such that $Pr(X=q\_n)\; =\; w\_n$, and $x\; =\; \backslash mathbb\; E[X]$.

Then, it is natural to consider the "variance" of a set $S$ as$$Var(S)\; :=\; \backslash sup\_\{x\; \backslash in\; \backslash operatorname\{conv\}S\}\; \backslash inf\_\{\backslash mathbb\; E[X]\; =\; x,\; X\backslash text\{\; is\; finitely\; supported\; in\; \}S\}\; Var[X]$$With that, $d(S,\; \backslash operatorname\{conv\}(S))^2\; \backslash leq\; Var(S)\; \backslash leq\; r(S)\; \backslash leq\; rad(S)$.{{Math proof|proof=

$d(S,\; \backslash operatorname\{conv\}(S))^2\; \backslash leq\; Var(S)$: Expand their definitions.

$Var(S)\; \backslash leq\; rad(S)$: if $x\backslash in\; \backslash operatorname\{conv\}(S)$ then let $X$ be finitely supported in $S$ such that $E[X]\; =\; x$. Now since $S$ is bounded in a ball of radius $rad(S)+\backslash epsilon$ centered at some $o$, we have $Var[X]\; =\; Var[X-o]\; \backslash leq\; E[\backslash |X-o\backslash |^2]\; \backslash leq\; rad(S)\; +\; \backslash epsilon$.

$Var(S)\; \backslash leq\; r(S)$: use the previous result.

}}{{Math proof|proof=

It suffices to show $Var\backslash left(Q\backslash right)\; \backslash leq\; \backslash sum\_\{\backslash max\; D\}\; Var(Q\_n)$.

$\backslash forall\; x\; \backslash in\; \backslash operatorname\{conv\}Q$, by Shapley–Folkman lemma, there exists a representation $x\; =\; \backslash sum\_\{n\backslash in\; I\}\; x\_n\; +\; \backslash sum\_\{n\; \backslash in\; J\}\; q\_n$, such that $I,\; J$ partitions $\backslash \{1,\; 2,\; ...,\; N\backslash \},\; |I|\; \backslash leq\; D,\; x\_n\; \backslash in\; \backslash operatorname\{conv\}Q\_n,\; q\_n\backslash in\; Q\_n$.

Now, for each $n\backslash in\; I$, construct random vectors $X\_n$ such that $X\_n$ is finitely supported on $Q\_n$, with $\backslash mathbb\; E[X\_n]\; =\; x\_n$ and , where $\backslash epsilon>\; 0$ is an arbitrary small number.

Let all such $X\_n$ be independent. Then let $X\; =\; \backslash sum\_\{n\backslash in\; I\}\; X\_n\; +\; \backslash sum\_\{n\backslash in\; J\}\; q\_n$. Since each $q\_n$ is a deterministic vector, we have

$$Var[X]\; =Var\; \backslash left[\backslash sum\_\{n\backslash in\; I\}\; X\_n\backslash right]=\; \backslash sum\_\{n\backslash in\; I\}\; Var\; \backslash left[\; X\_n\backslash right]\; \backslash leq\; \backslash sum\_\{n\backslash in\; I\}\backslash left(Var(Q\_n)\; +\; \backslash epsilon\backslash right)\; \backslash leq\; \backslash sum\_\{\backslash max\; D\}\; Var(Q\_n)\; +\; D\backslash epsilon.$$

Since this is true for arbitrary $\backslash epsilon\; >\; 0$, we have $Var[X]\; \backslash leq\; \backslash sum\_\{\backslash max\; D\}\; Var(Q\_n)$, and we are done.

|title=Proof of Shapley–Folkman–Starr theorem}}

File:Shapley, Lloyd (1923).jpeg proved the Shapley–Folkman lemma with Jon Folkman.]]The lemma of Lloyd Shapley and Jon Folkman was first{{sfnp|Zhou|1993}} published by the economist Ross M. Starr, who was investigating the existence of economic equilibria while studying with Kenneth Arrow. In his paper, Starr studied a convexified economy, in which non-convex sets were replaced by their convex hulls; Starr proved that the convexified economy has equilibria that are closely approximated by "quasi-equilibria" of the original economy; moreover, he proved that every quasi-equilibrium has many of the optimal properties of true equilibria, which are proved to exist for convex economies.

Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used to show that central results of (convex) economic theory are good approximations to large economies with non-convexities; for example, quasi-equilibria closely approximate equilibria of a convexified economy. "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Roger Guesnerie.{{harvtxt|Guesnerie|1989|p=138}}

The topic of non-convex sets in economics has been studied by many Nobel laureates: Shapley himself,{{sfnp|Shapley|Shubik|1966}} Arrow,{{harvtxt|Arrow|Hurwicz|1956}}. See p. 2: "The relaxation is primarily with regard to the convexity assumptions which fail to hold in some important economic applications". Robert Aumann,{{sfnp|Aumann|Hart|1986}} Gérard Debreu,{{sfnp|Debreu|Koopmans|1982}} Tjalling Koopmans,{{sfnp|Debreu|Koopmans|1982}} Paul Krugman,{{sfnp|Krugman|1992}} and Paul Samuelson.{{sfnp|Samuelson|1998}} The complementary topic of convex sets in economics has been emphasized by these laureates, along with Leonid Hurwicz, Leonid Kantorovich, and Robert Solow.

The Shapley–Folkman lemma enables researchers to extend results for Minkowski sums of convex sets to sums of general sets, which need not be convex. Such sums of sets arise in economics, in mathematical optimization, and in probability theory; in each of these three mathematical sciences, non-convexity is an important feature of applications.

File:Indifference curves showing budget line.svg every basket of goods on the indifference curve I₃ over each basket on  I₂.

The basket (Q_x, Q_y), where the budget line (shown in blue) supports I₂, is optimal and also feasible, unlike any basket lying on  I₃ which is preferred but unfeasible.]]

{{See also|Convexity in economics}}

In economics, a consumer's preferences are defined over all "baskets" of goods. Each basket is represented as a non-negative vector, whose coordinates represent the quantities of the goods. On this set of baskets, an indifference curve is defined for each consumer; a consumer's indifference curve contains all the baskets of commodities that the consumer regards as equivalent: That is, for every pair of baskets on the same indifference curve, the consumer does not prefer one basket over another. Through each basket of commodities passes one indifference curve. A consumer's preference set (relative to an indifference curve) is the union of the indifference curve and all the commodity baskets that the consumer prefers over the indifference curve. A consumer's preferences are convex if all such preference sets are convex.{{harvtxt|Mas-Colell|1985|pp=58–61}} and {{harvtxt|Arrow|Hahn|1980|pp=76–79}}

An optimal basket of goods occurs where the budget-line supports a consumer's preference set, as shown in the diagram. This means that an optimal basket is on the highest possible indifference curve given the budget-line, which is defined in terms of a price vector and the consumer's income (endowment vector). Thus, the set of optimal baskets is a function of the prices, and this function is called the consumer's demand. If the preference set is convex, then at every price the consumer's demand is a convex set, for example, a unique optimal basket or a line-segment of baskets.{{harvtxt|Arrow|Hahn|1980|pp=79–81}}

File:NonConvex.gif

{{See also|Non-convexity (economics)}}

However, if a preference set is non-convex, then some prices determine a budget-line that supports two separate optimal-baskets. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase half of an eagle and half of a lion (or a griffin)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.{{harvtxt|Starr|1969|p=26}}: "After all,

one may be indifferent between an automobile and a boat, but in most cases one can neither drive nor sail the combination of half boat, half car."

When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected; a disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling:

If indifference curves for purchases be thought of as possessing a wavy character, convex to the origin in some regions and concave in others, we are forced to the conclusion that it is only the portions convex to the origin that can be regarded as possessing any importance, since the others are essentially unobservable. They can be detected only by the discontinuities that may occur in demand with variation in price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight line is rotated. But, while such discontinuities may reveal the existence of chasms, they can never measure their depth. The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in unmeasurable obscurity.{{harvtxt|Hotelling|1935|p=74}}

The difficulties of studying non-convex preferences were emphasized by Herman Wold{{harvtxt|Wold|1943b|pp=231 and 239–240}} and {{harvtxt|Wold|1953|p=146}} and again by Paul Samuelson, who wrote that non-convexities are "shrouded in eternal darkness",{{harvtxt|Samuelson|1950|pp=359–360}}:

It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.

A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.

Nonetheless, non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in The Journal of Political Economy (JPE). The main contributors were Farrell,

{{harvs|txt|last=Farrell|year1=1959|year2=1961a|year3=1961b}} Bator,{{harvs|txt|last=Bator|year1=1961a|year2=1961b}} Koopmans,

{{harvtxt|Koopmans|1961|p=478}} and others—for example, {{harvtxt|Farrell|1959|pp=390–391}} and {{harvtxt|Farrell|1961a|p=484}}, {{harvtxt|Bator|1961a|pp=482–483}}, {{harvtxt|Rothenberg|1960|p=438}}, and {{harvtxt|Starr|1969|p=26}}—commented on {{harvtxt|Koopmans|1957|pp=1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]}} and Rothenberg.{{harvs|txt|last=Rothenberg|year1=1960|loc1=p. 447|year2=1961}}

In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets.{{harvtxt|Arrow|Hahn|1980|p=182}} These JPE-papers stimulated a paper by Lloyd Shapley and Martin Shubik, which considered convexified consumer-preferences and introduced the concept of an "approximate equilibrium".{{harvtxt|Shapley|Shubik|1966|p=806}} The JPE-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to Robert Aumann.{{harvtxt|Aumann|1966|pp=1–2}} uses results from

{{harvs|txt|last=Aumann|year1=1964|year2=1965}}Taking the convex hull of non-convex preferences had been discussed earlier by {{harvtxt|Wold|1943b|p=243}} and by {{harvtxt|Wold|1953|p=146}}, according to {{harvtxt|Diewert|1982|p=552}}.

File:Kenneth Arrow, Stanford University.jpg (1972 Nobel laureate) helped Ross M. Starr to study non-convex economies.]]

Previous publications on non-convexity and economics were collected in an annotated bibliography by Kenneth Arrow. He gave the bibliography to Starr, who was then an undergraduate enrolled in Arrow's (graduate) advanced mathematical-economics course.{{harvtxt|Starr|Stinchcombe|1999|pp=217–218}}

In his term-paper, Starr studied the general equilibria of an artificial economy in which non-convex preferences were replaced by their convex hulls. In the convexified economy, at each price, the aggregate demand was the sum of convex hulls of the consumers' demands. Starr's ideas interested the mathematicians Lloyd Shapley and Jon Folkman, who proved their eponymous lemma and theorem in "private correspondence", which was reported by Starr's published paper of 1969.

In his 1969 publication, Starr applied the Shapley–Folkman–Starr theorem. Starr proved that the "convexified" economy has general equilibria that can be closely approximated by "quasi-equilibria" of the original economy, when the number of agents exceeds the dimension of the goods: Concretely, Starr proved that there exists at least one quasi-equilibrium of prices $p\_\{\backslash mathrm\{opt\}\}$ with the following properties:

• For each quasi-equilibrium's prices $p\_\{\backslash mathrm\{opt\}\}$, all consumers can choose optimal baskets (maximally preferred and meeting their budget constraints).
• At quasi-equilibrium prices $p\_\{\backslash mathrm\{opt\}\}$ in the convexified economy, every good's market is in equilibrium: Its supply equals its demand.
• For each quasi-equilibrium, the prices "nearly clear" the markets for the original economy: an upper bound on the distance between the set of equilibria of the "convexified" economy and the set of quasi-equilibria of the original economy followed from Starr's corollary to the Shapley–Folkman theorem.{{harvtxt|Arrow|Hahn|1980|pp=169–182}} and {{harvtxt|Starr|1969|pp=27–33}}

Starr established that

"in the aggregate, the discrepancy between an allocation in the fictitious economy generated by [taking the convex hulls of all of the consumption and production sets] and some allocation in the real economy is bounded in a way that is independent of the number of economic agents. Therefore, the average agent experiences a deviation from intended actions that vanishes in significance as the number of agents goes to infinity".{{harvtxt|Green|Heller|1981|p=44}}

Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used in economic theory. Roger Guesnerie summarized their economic implications: "Some key results obtained under the convexity assumption remain (approximately) relevant in circumstances where convexity fails. For example, in economies with a large consumption side, preference nonconvexities do not destroy the standard results".{{harvtxt|Guesnerie|1989|pp=99}} "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Guesnerie. The topic of non-convex sets in economics has been studied by many Nobel laureates: Arrow (1972), Robert Aumann (2005), Gérard Debreu (1983), Tjalling Koopmans (1975), Paul Krugman (2008), and Paul Samuelson (1970); the complementary topic of convex sets in economics has been emphasized by these laureates, along with Leonid Hurwicz, Leonid Kantorovich (1975), and Robert Solow (1987).{{harvtxt|Mas-Colell|1987}} The Shapley–Folkman–Starr results have been featured in the economics literature: in microeconomics,{{harvtxt|Varian|1992|pp=393–394}}

{{harvtxt|Mas-Colell|Whinston|Green|1995|pp=627–630}}

File:Epigraph convex.svg is convex if the region above its graph is a convex set.]]

The Shapley–Folkman lemma has been used to explain why large minimization problems with non-convexities can be nearly solved (with iterative methods whose convergence proofs are stated for only convex problems). The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.

Nonlinear optimization relies on the following definitions for functions:

• The graph of a function f is the set of the pairs of arguments x and function evaluations f(x)

: Graph(f) = { (x, f(x) ) }

• The epigraph of a real-valued function f is the set of points above the graph

File:Sine.svg is non-convex.]]

: Epi(f) = { (x, u) : f(x) ≤ u }.

• A real-valued function is defined to be a convex function if its epigraph is a convex set.{{harvtxt|Rockafellar|1997|p=23}}

For example, the quadratic function f(x) = x² is convex, as is the absolute value function g(x) = |x|. However, the sine function (pictured) is non-convex on the interval (0, π).

In many optimization problems, the objective function f is separable: that is, f is the sum of many summand-functions, each of which has its own argument:

$$f(x)=f\backslash bigl((x\_1,\backslash dots,x\_N)\backslash bigr)=\backslash sum\; f\_n(x\_n).$$

For example, problems of linear optimization are separable. Given a separable problem with an optimal solution, fix an optimal solution

$$x\_\{\backslash min\}=(x\_1,\backslash dots,x\_N)\_\{\backslash min\}$$

with the minimum value $f(x\_\{\backslash min\})$. For this separable problem, consider an optimal solution $\backslash bigl(x\_\{\backslash min\},f(x\_\{\backslash min\})\backslash bigr)$ to the convexified problem, where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the limit of a sequence{{efn|1=

The limit of a sequence is a member of the closure of the original set, which is the smallest closed set that contains the original set. The Minkowski sum of two closed sets need not be closed, so the following inclusion (where $\backslash operatorname\{cl\}$ denotes the topological closure operator) can be strict:

$$\backslash operatorname\{cl\}P+\backslash operatorname\{cl\}Q\backslash subseteq\; \backslash operatorname\{cl\}(\backslash operatorname\{cl\}P+\backslash operatorname\{cl\}Q).$$

The inclusion can be strict even for two convex closed summand-sets.{{sfnp|Rockafellar|1997|pp=49 and 75}} Ensuring that the Minkowski sum of sets be closed requires the closure operation, which appends limits of convergent sequences.}} of points in the convexified problem

$$\backslash bigl(x\_j,f(x\_j)\backslash bigr)\backslash in\backslash sum\backslash operatorname\{conv\}\; \backslash operatorname\{Graph\}(f\_n).$$

The given optimal-point is a sum of points in the graphs of the original summands and of a small number of convexified summands, by the Shapley–Folkman lemma.

This analysis was published by Ivar Ekeland in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician Claude Lemaréchal was surprised by his success with convex minimization methods on problems that were known to be non-convex; for minimizing nonlinear problems, a solution of the dual problem need not provide useful information for solving the primal problem, unless the primal problem be convex and satisfy a constraint qualification. Lemaréchal's problem was additively separable, and each summand function was non-convex; nonetheless, a solution to the dual problem provided a close approximation to the primal problem's optimal value.{{harvtxt|Lemaréchal|1973|p=38}}

Lemaréchal's experiments were discussed in later publications:

{{harvtxt|Aardal|1995|pp=2–3}}

{{harvtxt|Hiriart-Urruty|Lemaréchal|1993|pp=143–145, 151, 153, and 156}}

Convex sets are often studied with probability theory. Each point in the convex hull of a (non-empty) subset Q of a finite-dimensional space is the expected value of a simple random vector that takes its values in Q, as a consequence of Carathéodory's lemma. Thus, for a non-empty set Q, the collection of the expected values of the simple, Q-valued random vectors equals Q{{'s}} convex hull; this equality implies that the Shapley–Folkman–Starr results are useful in probability theory.{{harvtxt|Schneider|Weil|2008|p=45}} In the other direction, probability theory provides tools to examine convex sets generally and the Shapley–Folkman–Starr results specifically.{{harvtxt|Cassels|1975|pp=433–434}}

The Shapley–Folkman–Starr results have been widely used in the probabilistic theory of random sets,{{harvtxt|Molchanov|2005|pp=195–198, 218, 232, 237–238 and 407}} for example, to prove a law of large numbers,{{harvtxt|Artstein|Vitale|1975|pp=881–882}}{{harvtxt|Puri|Ralescu|1985|pp=154–155}}

a central limit theorem,{{harvtxt|Weil|1982|pp=203, and 205–206}} and a large-deviations principle.{{harvtxt|Cerf|1999|pp=243–244}} uses applications of the Shapley–Folkman lemma from {{harvtxt|Puri|Ralescu|1985|pp=154–155}}. These proofs of probabilistic limit theorems used the Shapley–Folkman–Starr results to avoid the assumption that all the random sets be convex.{{cn|date=May 2025}}

A probability measure is a finite measure, and the Shapley–Folkman lemma has applications in non-probabilistic measure theory, such as the theories of volume and of vector measures. The Shapley–Folkman lemma enables a refinement of the Brunn–Minkowski inequality, which bounds the volume of sums in terms of the volumes of their summand-sets.{{harvtxt|Ruzsa|1997|p=345}} The volume of a set is defined in terms of the Lebesgue measure, which is defined on subsets of Euclidean space. In advanced measure-theory, the Shapley–Folkman lemma has been used to prove Lyapunov's theorem, which states that the range of a vector measure is convex.{{harvtxt|Tardella|1990|pp=478–479}} Here, the traditional term "range" (alternatively, "image") is the set of values produced by the function.

A vector measure is a vector-valued generalization of a measure;

for example,

if p₁ and p₂ are probability measures defined on the same measurable space,

then the product function {{nowrap|p₁ p₂}} is a vector measure,

where {{nowrap|p₁ p₂}}

is defined for every event ω

by

:(p₁ p₂)(ω)=(p₁(ω), p₂(ω)).

Lyapunov's theorem has been used in economics,{{harvtxt|Vind|1964|pp=168 and 175}} was noted by the winner of the 1983 Nobel Prize in Economics, Gérard Debreu. {{harvtxt|Debreu|1991|p=4}} wrote:

The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of economic competition: If one associates with every agent of an economy an arbitrary set in the commodity space and if one averages those individual sets over a collection of insignificant agents, then the resulting set is necessarily convex. [Debreu appends this footnote: "On this direct consequence of a theorem of A. A. Lyapunov, see {{harvtxt|Vind|1964}}."] But explanations of the ... functions of prices ... can be made to rest on the convexity of sets derived by that averaging process. Convexity in the commodity space obtained by aggregation over a collection of insignificant agents is an insight that economic theory owes ... to integration theory. [Italics added]

in ("bang-bang") control theory, and in statistical theory.{{harvtxt|Artstein|1980|pp=172–183}} Lyapunov's theorem has been called a continuous counterpart of the Shapley–Folkman lemma, which has itself been called a discrete analogue of Lyapunov's theorem.{{harvtxt|Mas-Colell|1978|p=210}}

{{clear}}

{{notelist}}

{{reflist}}

{{refbegin|30em}}

• {{cite journal|last=Aardal|first=Karen|author-link=Karen Aardal|title=Optima interview - Claude Lemaréchal|journal=Optima: Mathematical Programming Society Newsletter|pages=2–4|date=March 1995|volume=45 |url=http://www.mathprog.org/Old-Optima-Issues/optima45.pdf|access-date=2 February 2011}}
• {{cite book|title=Economics 201B: Nonconvex preferences and approximate equilibria|chapter=1 The Shapley–Folkman theorem|pages=1–5|date=14 March 2005|first=Robert M.|last=Anderson|author-link=Robert M. Anderson (mathematician)|location=Berkeley, Calif.|publisher=Economics Department, University of California, Berkeley|chapter-url=https://eml.berkeley.edu//~anderson/Econ201B/NonconvexHandout.pdf|access-date=2025-06-03}}
• {{cite book|last1=Arrow|first1=Kenneth J.|author-link1=Kenneth Arrow|last2=Hahn|first2=Frank H. |author-link2=Frank Hahn|year=1980|title=General competitive analysis|publisher=North-Holland|series=Advanced Textbooks in Economics|volume=12|edition=reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6|orig-year=1971|location=Amsterdam|isbn=0-444-85497-5|mr=439057}}
• {{cite conference

| last1 = Arrow | first1 = Kenneth J. | author1-link = Kenneth Arrow

| last2 = Hurwicz | first2 = Leonid

| contribution = Reduction of constrained maxima to saddle-point problems

| contribution-url = https://digicoll.lib.berkeley.edu/record/112870/files/math_s3_v5_article-01.pdf

| mr = 84938

| pages = 1–20

| publisher = Univ. California Press, Berkeley-Los Angeles, Calif.

| title = Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. V

| year = 1956}}

• {{cite journal|last=Artstein|first=Zvi|title=Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points|journal=SIAM Review|volume=22|year=1980|issue=2|pages=172–185 |doi=10.1137/1022026 |mr=564562|jstor=2029960}} Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
• {{cite book|ref=none|last=Artstein|first=Zvi|chapter=22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points|pages=449–462|title=Game and economic theory: Selected contributions in honor of Robert J. Aumann|chapter-url=http://www.press.umich.edu/titleDetailDesc.do?id=14414 |editor1-first=Sergiu|editor1-last=Hart|editor2-first=Abraham|editor2-last=Neyman |publisher=University of Michigan Press|location=Ann Arbor, Mich. |year=1995|isbn=0-472-10673-2|url-status=dead|archive-date=24 May 2011 |archive-url=https://web.archive.org/web/20110524002010/http://press.umich.edu/titleDetailDesc.do?id=14414}}
• {{cite journal|last1=Artstein|first1=Zvi|last2=Vitale|first2=Richard A.|year=1975|title=A strong law of large numbers for random compact sets|journal=The Annals of Probability|volume=3|issue=5|pages=879–882 |doi=10.1214/aop/1176996275|mr=385966|jstor=2959130|zbl=0313.60012|id={{Project Euclid|euclid.ss/1176996275}}|doi-access=free}}
• {{cite book|last=Aubin|first=Jean-Pierre|chapter=14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The Shapley–Folkman theorem, pages 463–465)|title=Mathematical methods of game and economic theory|edition=Reprint with new preface of 1982 North-Holland revised English|publisher=Dover Publications|location=Mineola, NY|year=2007|isbn=978-0-486-46265-3|mr=2449499}}
• {{cite journal|last1=Aubin|first1=J. P.|last2=Ekeland|first2=I.|author2-link=Ivar Ekeland|journal=Mathematics of Operations Research|title=Estimates of the duality gap in nonconvex optimization|volume=1|issue=3|pages=225–245 |year=1976 |doi=10.1287/moor.1.3.225|mr=449695|jstor=3689565}}
• {{cite journal|last=Aumann|first=Robert J.|author-link=Robert Aumann|title=Markets with a continuum of traders|journal=Econometrica|volume=32|issue=1–2|date=January–April 1964|pages=39–50|jstor=1913732 |mr=172689 |doi=10.2307/1913732}}
• {{cite journal|last=Aumann|first=Robert J.|author-link=Robert Aumann|title=Integrals of set-valued functions|journal=Journal of Mathematical Analysis and Applications|volume=12|issue=1|date=August 1965 |pages=1–12 |doi=10.1016/0022-247X(65)90049-1|mr=185073|doi-access=free}}
• {{cite journal|last=Aumann|first=Robert J.|author-link=Robert Aumann|title=Existence of competitive equilibrium in markets with a continuum of traders|journal=Econometrica|volume=34|issue=1|date=January 1966|pages=1–17 |jstor=1909854|mr=191623|doi=10.2307/1909854}}
• {{cite journal

| last1 = Aumann | first1 = Robert J.|author1-link=Robert Aumann

| last2 = Hart | first2 = Sergiu|author2-link = Sergiu Hart

| doi = 10.1007/BF02764940

| issue = 2

| journal = Israel Journal of Mathematics

| mr = 852476

| pages = 159–180

| title = Bi-convexity and bi-martingales

| volume = 54

| year = 1986}}

• {{cite book

| last1 = Balestro | first1 = Vitor

| last2 = Martini | first2 = Horst

| last3 = Teixeira | first3 = Ralph

| doi = 10.1007/978-3-031-50507-2

| isbn = 978-3-031-50506-5

| location = Cham

| mr = 4830872

| publisher = Birkhäuser/Springer

| series = Cornerstones

| title = Convexity from the Geometric Point of View

| year = 2024}}

• {{cite journal|last=Bator|first=Francis M. |author-link=Francis M. Bator|title=On convexity, efficiency, and markets|journal=Journal of Political Economy|volume=69|issue=5|date=October 1961a|pages=480–483|jstor=1828537 |doi=10.1086/258540}}
• {{cite journal|last=Bator|first=Francis M. |author-link=Francis M. Bator|title=On convexity, efficiency, and markets: Rejoinder|journal=Journal of Political Economy|volume=69|issue=5|date=October 1961b|pages=489 |jstor=1828539 |doi=10.1086/258542}}
• {{cite book|last=Bertsekas|first=Dimitri P.|author-link=Dimitri P. Bertsekas|title=Nonlinear Programming|edition=Second|chapter=5.1.6 Separable problems and their geometry|pages=494–498|publisher=Athena Scientific|year=1999 |location=Cambridge, Mass.|isbn =1-886529-00-0}}
• {{cite book|last=Bertsekas|first=Dimitri P.|author-link=Dimitri P. Bertsekas|chapter=5.6 Large scale separable integer programming problems and the exponential method of multipliers|title=Constrained optimization and Lagrange multiplier methods|year=1996|location=Belmont, Mass. |isbn=1-886529-04-3|publisher=Athena Scientific|mr=690767}} Reprint of (1982) Academic Press.
• {{cite book

| last = Bertsekas

| first = Dimitri P.

| author-link = Dimitri P. Bertsekas

| title = Convex Optimization Theory

| publisher = Athena Scientific

| year = 2009

| location = Belmont, Mass.

| isbn = 978-1-886529-31-1

}}

• {{cite journal|last1=Bertsekas|first1=Dimitri P.|author-link1=Dimitri Bertsekas|last2=Lauer |first2=Gregory S.|last3=Sandell|first3=Nils R. Jr. |last4=Posbergh|first4=Thomas A.|journal=IEEE Transactions on Automatic Control |volume=28|date=January 1983|title=Optimal short-term scheduling of large-scale power systems|pages=1–11|doi=10.1109/tac.1983.1103136|issue=1 |url=http://web.mit.edu/dimitrib/www/Unit_Comm.pdf|access-date=2 February 2011}} Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
• {{cite journal

| last = Blumson | first = Ben

| doi = 10.1111/1746-8361.12263

| issue = 1-2

| journal = Dialectica

| mr = 3994693

| pages = 65–81

| title = Naturalness and convex class nominalism

| volume = 73

| year = 2019}}

• {{cite journal|last1=Borwein|first1=J. M.|author-link=Jonathan Borwein|last2=O'Brien|first2=R. C. |title=Cancellation characterizes convexity|journal=Nanta Mathematica (Nanyang University) |issn=0077-2739 |volume=11|year=1978 |pages=100–102|mr=510842}}
• {{cite book|title=Fundamentals of Mathematical Analysis|first=Adel N.|last=Boules|publisher=Oxford University Press|year=2021|isbn=9780192639448}}
• {{cite book|last=Carter|first=Michael|title=Foundations of Mathematical Economics|publisher=MIT Press |location=Cambridge, Mass.|year=2001|isbn=0-262-53192-5|mr=1865841}}
• {{cite journal|last=Cassels|first=J. W. S.|author-link=J. W. S. Cassels|title=Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem|journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=78|year=1975|issue=3|pages=433–436|doi=10.1017/S0305004100051884|mr=385711}}
• {{cite book|last=Cassels|first=J. W. S.|author-link=J. W. S. Cassels|chapter=Appendix A: Convex sets|title=Economics for mathematicians |series=London Mathematical Society lecture note series |volume=62 |publisher=Cambridge University Press|location=Cambridge, UK|year=1981|isbn=0-521-28614-X |mr=657578}}
• {{cite journal|last=Cerf|first=Raphaël|author-link=Raphaël Cerf|title=Large deviations for sums of i.i.d. random compact sets |journal=Proceedings of the American Mathematical Society|volume=127|year=1999|issue=8|pages=2431–2436|doi=10.1090/S0002-9939-99-04788-7|mr=1487361|doi-access=free}}
• {{cite journal|last=Debreu|first=Gérard|author-link=Gérard Debreu|title=The Mathematization of economic theory|issue=Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC|journal=The American Economic Review|volume=81|date=March 1991|pages=1–7|jstor=2006785}}
• {{cite journal

| last1 = Debreu | first1 = Gerard |author1-link=Gérard Debreu

| last2 = Koopmans | first2 = Tjalling C. | author2-link = Tjalling Koopmans

| doi = 10.1007/BF01585092

| issue = 1

| journal = Mathematical Programming

| mr = 667937

| pages = 1–38

| title = Additively decomposed quasiconvex functions

| volume = 24

| year = 1982}}

• {{cite journal|last=Di Guglielmo|first=F.|title=Nonconvex duality in multiobjective optimization |doi=10.1287/moor.2.3.285|volume=2|year=1977|issue=3|pages=285–291|journal=Mathematics of Operations Research |mr=484418 |jstor=3689518}}
• {{cite book|first=W. E.|last=Diewert|chapter=12: Duality approaches to microeconomic theory

|pages=535–599

|chapter-url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-R/2/dcc0f8c9352eb054c96b3ff481976ce7

|doi=10.1016/S1573-4382(82)02007-4

|title=Handbook of Mathematical Economics|editor1-link=Kenneth Arrow |editor1-first=Kenneth Joseph|editor1-last=Arrow|editor2-first=Michael D.|editor2-last=Intriligator|editor2-link=Michael Intriligator|series=Handbooks in Economics|volume=2 |publisher=North-Holland Publishing|location=Amsterdam|year=1982|isbn=978-0-444-86127-6|mr=648778}}

• {{cite journal|last=Ekeland|first=Ivar|author-link=Ivar Ekeland |title=Une estimation a priori en programmation non convexe|journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences |series=Séries A et B|language=fr|issn=0151-0509|volume=279|year=1974|pages=149–151|mr=395844}}
• {{cite book|last=Ekeland|first=Ivar|author-link=Ivar Ekeland|chapter=Appendix I: An a priori estimate in convex programming|editor1-last=Ekeland|editor1-first=Ivar|editor2-last=Temam|editor2-first=Roger|editor2-link=Roger Temam|title=Convex analysis and variational problems |edition=Corrected reprinting of the North-Holland|orig-year=1976|series=Classics in Applied Mathematics |volume=28|publisher=Society for Industrial and Applied Mathematics (SIAM)|location=Philadelphia |year=1999|pages=357–373|isbn=0-89871-450-8|mr=1727362}}
• {{cite book|last=Ellickson|first=Bryan |title=Competitive equilibrium: Theory and applications |publisher=Cambridge University Press|isbn=978-0-521-31988-1|year=1994|doi = 10.1017/CBO9780511609411}}
• {{cite journal|last=Farrell|first=M. J.|author-link=Michael James Farrell|date=August 1959|title=The Convexity assumption in the theory of competitive markets|journal=Journal of Political Economy|volume=67|issue=4|pages=371–391|jstor=1825163 |doi=10.1086/258197}}
• {{cite journal|last=Farrell|first=M. J.|author-link=Michael James Farrell|date=October 1961a|title=On Convexity, efficiency, and markets: A Reply|journal=Journal of Political Economy|volume=69|issue=5|pages=484–489|jstor=1828538 |doi=10.1086/258541}}
• {{cite journal|last=Farrell|first=M. J.|author-link=Michael James Farrell|date=October 1961b|title=The Convexity assumption in the theory of competitive markets: Rejoinder|journal=Journal of Political Economy|volume=69|issue=5 |pages=493|jstor=1828541 |doi=10.1086/258544}}
• {{cite book|first1=Jerry|last1=Green|author1-link=Jerry Green (economist)|first2=Walter P.|last2=Heller|author2-link=Walter Heller|chapter=1 Mathematical analysis and convexity with applications to economics|pages=15–52|doi=10.1016/S1573-4382(81)01005-9|title=Handbook of mathematical economics|editor1-link=Kenneth Arrow |editor1-first=Kenneth Joseph|editor1-last=Arrow|editor2-first=Michael D. |editor2-last=Intriligator|series=Handbooks in Economics|volume=1 |publisher=North-Holland Publishing |location=Amsterdam|year=1981|isbn=0-444-86126-2|mr=634800}}
• {{cite book|last=Guesnerie|first=Roger|author-link=Roger Guesnerie|year=1989|chapter=First-best allocation of resources with nonconvexities in production|pages=99–143|editor-first=Bernard|editor-last=Cornet|editor2-first=Henry|editor2-last=Tulkens|title=Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987)|publisher=MIT Press |location=Cambridge, Mass.|isbn=0-262-03149-3|mr=1104662}}
• {{cite book|last=Hildenbrand|first=Werner|author-link=Werner Hildenbrand|title=Core and equilibria of a large economy|series=Princeton studies in mathematical economics|volume=5|publisher=Princeton University Press |location=Princeton, NJ|year=1974|isbn=978-0-691-04189-6|mr=389160}}
• {{cite book|last1=Hiriart-Urruty|first1=Jean-Baptiste|last2=Lemaréchal|first2=Claude|author-link2=Claude Lemaréchal|year=1993|chapter=XII Abstract duality for practitioners|title=Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods|series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]|volume=306|publisher=Springer-Verlag |location=Berlin |pages=136–193 (and bibliographical comments on pp. 334–335)|isbn=3-540-56852-2 |mr=1295240}}
• {{cite journal|first=Harold|last=Hotelling|author-link=Harold Hotelling |title=Demand functions with limited budgets|journal=Econometrica|volume=3|issue=1|date=January 1935|pages=66–78|jstor=1907346 |doi=10.2307/1907346}}
• {{cite report|last=Howe|first=Roger|author-link=Roger Evans Howe|date=November 1979|title=On the tendency toward convexity of the vector sum of sets |publisher=Cowles Foundation for Research in Economics, Yale University|series=Cowles Foundation discussion papers|location=New Haven, Conn.|volume=538 |url=http://cowles.econ.yale.edu/P/cd/d05a/d0538.pdf |access-date=15 January 2011}}
• {{cite book|last=Ichiishi|first=Tatsuro |title=Game Theory for Economic Analysis|series=Economic theory, econometrics, and mathematical economics|publisher=Academic Press [Harcourt Brace Jovanovich, Publishers] |location=New York |year=1983|isbn=0-12-370180-5|mr=700688}}
• {{cite book|last=Koopmans|first=Tjalling C.|author-link=Tjalling Koopmans|year=1957 |chapter=Allocation of resources and the price system|editor-last=Koopmans|editor-first=Tjalling C |editor-link=Tjalling Koopmans |title=Three essays on the state of economic science|publisher=McGraw–Hill Book Company|location=New York|pages=1–126|isbn=((0-07-035337-9))}}
• {{cite journal|last=Koopmans |first=Tjalling C. |author-link=Tjalling Koopmans |date=October 1961 |title=Convexity assumptions, allocative efficiency, and competitive equilibrium |journal=Journal of Political Economy |volume=69 |issue=5 |pages=478–479 |jstor=1828536 |doi=10.1086/258539}}
• {{cite journal

| last = Krugman | first = Paul | author-link = Paul Krugman

| date = July 1992

| doi = 10.1111/j.1467-9701.1992.tb00528.x

| issue = 4

| journal = The World Economy

| pages = 423–442

| title = Does the New Trade Theory Require a New Trade Policy?

| volume = 15}}

• {{cite book |last=Laffont|first=Jean-Jacques|author-link=Jean-Jacques Laffont|year=1988 |chapter=3. Nonconvexities|title=Fundamentals of Public Economics|publisher=MIT Press|pp=63–65|isbn=0-262-12127-1 }}
• {{cite report|last=Lemaréchal|first=Claude|author-link=Claude Lemaréchal|title=Utilisation de la dualité dans les problémes non convexes [Use of duality for non–convex problems]|language=fr| date=April 1973 |issue=16|location=Domaine de Voluceau, Rocquencourt, Le Chesnay, France|publisher=IRIA (now INRIA), Laboratoire de recherche en informatique et automatique}}
• {{cite journal|last=Mas-Colell|first=Andreu|author-link=Andreu Mas-Colell|year=1978|title=A note on the core equivalence theorem: How many blocking coalitions are there?|journal=Journal of Mathematical Economics |volume=5 |issue=3|pages=207–215|doi=10.1016/0304-4068(78)90010-1|mr=514468}}
• {{cite book|last=Mas-Colell|first=Andreu|author-link=Andreu Mas-Colell|year=1985|chapter=1.L Averages of sets|title=The Theory of General Economic Equilibrium: A Differentiable Approach|series=Econometric Society monographs |volume=9|publisher=Cambridge University Press|isbn=0-521-26514-2|mr=1113262}}
• {{cite book|last=Mas-Colell|first=A.|author-link=Andreu Mas-Colell|chapter=Non-convexity|title=The New Palgrave: A Dictionary of Economics|editor1-first=John|editor1-last=Eatwell|editor1-link=John Eatwell, Baron Eatwell|editor2-first=Murray|editor2-last=Milgate|editor2-link=Murray Milgate|editor3-first=Peter|editor3-last=Newman|editor3-link=Peter Kenneth Newman|publisher=Palgrave Macmillan|year=1987|edition=first|doi=10.1057/9780230226203.3173|pages=653–661|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde1987_X001573|id=([http://www.econ.upf.edu/~mcolell/research/art_083b.pdf PDF file at Mas-Colell's homepage])|title-link=The New Palgrave Dictionary of Economics|isbn=9780333786765}}
• {{cite book|last1=Mas-Colell|first1=Andreu|author-link=Andreu Mas-Colell|last2=Whinston|first2=Michael D. |first3=Jerry R.|last3=Green|chapter=17.1 Large economies and nonconvexities |title=Microeconomic Theory |publisher=Oxford University Press|year=1995|isbn=978-0-19-507340-9}}
• {{cite book|last=Molchanov|first=Ilya|chapter=3: Minkowski addition|title=Theory of random sets |series=Probability and its Applications |publisher=Springer-Verlag London|location=London|year=2005 |isbn=978-1-84996-949-9 |pages=[https://archive.org/details/theoryrandomsets00molc_724/page/n209 194]–240 |doi=10.1007/1-84628-150-4|mr=2132405 |url=https://archive.org/details/theoryrandomsets00molc_724|url-access=limited}}
• {{cite book|title=Metric Spaces, Convexity and Nonpositive Curvature|volume=6|series=IRMA lectures in mathematics and theoretical physics|first=Athanase|last=Papadopoulos|publisher=European Mathematical Society|year=2005|isbn=9783037190104}}
• {{cite journal|last1=Puri|first1=Madan L.|author1-link=Madan Lal Puri|last2=Ralescu|first2=Dan A.|title=Limit theorems for random compact sets in Banach space|journal=Mathematical Proceedings of the Cambridge Philosophical Society |year=1985 |volume=97|issue=1|pages=151–158|doi=10.1017/S0305004100062691|mr=764504 |bibcode=1985MPCPS..97..151P}}
• {{cite book|last=Rockafellar|first=R. Tyrrell|author-link=R. Tyrrell Rockafellar|title=Convex Analysis|series=Princeton Landmarks in Mathematics|publisher=Princeton University Press|location=Princeton, NJ |year=1997 |isbn=0-691-01586-4|mr=1451876}}. Reprint of the 1970 ({{MR|274683}}) Princeton Mathematical Series 28
• {{cite journal |last=Rothenberg |first=Jerome |date=October 1960 |title=Non-convexity, aggregation, and Pareto optimality |journal=Journal of Political Economy |volume=68 |issue=5 |pages=435–468 |jstor=1830308 |doi=10.1086/258363}}
• {{cite journal |last=Rothenberg |first=Jerome |date=October 1961 |title=Comments on non-convexity |journal=Journal of Political Economy |volume=69 |issue=5 |pages=490–492 |jstor=1828540|doi=10.1086/258543}}
• {{cite journal|last=Ruzsa|first=Imre Z.|author-link=Imre Z. Ruzsa|title=The Brunn–Minkowski inequality and nonconvex sets|journal=Geometriae Dedicata|volume=67|doi=10.1023/A:1004958110076 |doi-access=free |year=1997|issue=3|pages=337–348 |mr=1475877}}
• {{cite book|last=Salanié|first=Bernard|author-link=Bernard Salanié|chapter=7 Nonconvexities |title=Microeconomics of Market Failures |year=2000|publisher=MIT Press|location=Cambridge, Mass.|pages=107–125|isbn=0-262-19443-0}} English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris)
• {{cite journal|last=Samuelson|first=Paul A.|author-link=Paul Samuelson|title=The problem of integrability in utility theory|journal=Economica|series=New Series|volume=17 |issue=68|date=November 1950|pages=355–385 |mr=43436 |jstor=2549499|doi=10.2307/2549499}}
• {{cite journal

| last = Samuelson | first = Paul A. | author-link = Paul Samuelson

| date = September 1998

| issue = 3

| journal = Journal of Economic Literature

| jstor = 2564803

| pages = 1375–1386

| title = How foundations came to be

| volume = 36}}

• {{cite journal|last1=Shapley|first1=L. S.|author-link1=Lloyd Shapley|last2=Shubik|first2=M.|author-link2=Martin Shubik|title=Quasi-cores in a monetary economy with nonconvex preferences|journal=Econometrica |volume=34 |issue=4|date=October 1966|pages=805–827|jstor=1910101|zbl=0154.45303|doi=10.2307/1910101 |url=http://www.dtic.mil/get-tr-doc/pdf?AD=AD0623552|archive-url=https://web.archive.org/web/20170924084740/http://www.dtic.mil/get-tr-doc/pdf?AD=AD0623552|url-status=dead|archive-date=24 September 2017}}
• {{cite book|last=Schneider|first=Rolf|author-link=Rolf Schneider|title=Convex Bodies: The Brunn–Minkowski Theory |series=Encyclopedia of Mathematics and its Applications|volume=44|publisher=Cambridge University Press |location=Cambridge, UK|year=1993 |isbn=0-521-35220-7|mr=1216521 |url=https://archive.org/details/convexbodiesbrun0000schn}}
• {{cite book|last1=Schneider|first1=Rolf|author1-link=Rolf Schneider|last2=Weil|first2=Wolfgang|title=Stochastic and Integral Geometry |series=Probability and its applications|doi=10.1007/978-3-540-78859-1|year=2008|publisher=Springer |isbn=978-3-540-78858-4 |mr=2455326}}
• {{cite journal|last=Starr|first=Ross M.|author-link=Ross Starr|issue=1|journal=Econometrica|pages=25–38|title=Quasi-equilibria in markets with non-convex preferences|volume=37|year=1969|jstor=1909201|doi=10.2307/1909201}} Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
• {{cite journal

| last = Starr | first = Ross M.|author-link=Ross Starr

| date = October 1981

| doi = 10.1016/0022-0531(81)90010-7

| issue = 2

| journal = Journal of Economic Theory

| mr = 640201

| pages = 314–317

| title = Approximation of points of the convex hull of a sum of sets by points of the sum: An elementary approach

| url = https://econweb.ucsd.edu/~rstarr/ElementarySum.pdf

| volume = 25}}

• {{cite book|last=Starr |first=Ross M.|author-link=Ross Starr|chapter=8 Convex sets, separation theorems, and non-convex sets in R ^$N$|title=General Equilibrium Theory: An Introduction |edition=1st|publisher=Cambridge University Press|location=Cambridge, UK |year=1997|isbn=0-521-56473-5|mr=1462618}} New chapters 22 and 25–26 in (2011) second ed.
• {{cite book|last=Starr|first=Ross M.|chapter=Shapley–Folkman theorem|title=The new Palgrave Dictionary of Economics|editor-first=Steven N.|editor-last=Durlauf|editor2-first=Lawrence E.|editor2-last=Blume|editor1-link=Steven N. Durlauf|editor2-link=Lawrence E. Blume|publisher=Palgrave Macmillan|year=2008|edition=Second|pages=317–318 (1st ed.)|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_S000107 |doi=10.1057/9780230226203.1518 |title-link=The New Palgrave Dictionary of Economics|isbn=978-0-333-78676-5}}
• {{cite book|last1=Starr|first1=Ross M.|author-link1=Ross Starr|last2=Stinchcombe|first2=M. B. |chapter=Exchange in a network of trading posts|title=Markets, Information and Uncertainty: Essays in Economic Theory in Honor of Kenneth J. Arrow|editor-first=Graciela|editor-last=Chichilnisky|editor-link=Graciela Chichilnisky|pages=217–234|publisher=Cambridge University Press|location=Cambridge, UK |year=1999|isbn=978-0-521-08288-4|doi = 10.1017/CBO9780511896583}}
• {{cite book|title=Mathematical Economics|first=Akira|last=Takayama|publisher=Cambridge University Press|year=1985|isbn=9780521314985}}
• {{cite journal|last=Tardella|first=Fabio|title=A new proof of the Lyapunov convexity theorem|journal=SIAM Journal on Control and Optimization|volume=28|year=1990|issue=2|pages=478–481 |doi=10.1137/0328026 |mr=1040471}}
• {{cite book|last=Trockel|first=Walter|title=Market Demand: An Analysis of Large Economies with Nonconvex Preferences|series=Lecture Notes in Economics and Mathematical Systems|volume=223|publisher=Springer-Verlag |location=Berlin |year=1984|isbn=3-540-12881-6|mr=737006}}
• {{cite book|last=Varian|first=Hal R.|author-link=Hal Varian|chapter=21.2 Convexity and size |title=Microeconomic Analysis|publisher=W. W. Norton & Company|edition=3rd|year=1992|isbn=978-0-393-95735-8 |mr=1036734 |url-access=registration|url=https://archive.org/details/microeconomicana00vari_0}}
• {{cite journal|last=Vind|first=Karl|date=May 1964|title=Edgeworth-allocations in an exchange economy with many traders|journal=International Economic Review|volume=5|pages=165–177|issue=2|jstor=2525560 |doi=10.2307/2525560}}
• {{cite journal|last=Weil|first=Wolfgang|title=An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets|journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete |trans-journal=Probability Theory and Related Fields|volume=60|year=1982 |issue=2|pages=203–208|doi=10.1007/BF00531823|mr=663901|doi-access=free}}
• {{cite journal|last=Wold|first=Herman|author-link=Herman Wold|year=1943b|title=A synthesis of pure demand analysis II|journal=Skandinavisk Aktuarietidskrift |trans-journal=Scandinavian Actuarial Journal|volume=26 |pages=220–263|mr=11939|doi=10.1080/03461238.1943.10404737}}
• {{cite book|last1=Wold|first1=Herman|author-link1=Herman Wold|others=In association with Lars Juréen|chapter=8 Some further applications of preference fields (pp. 129–148)|title=Demand Analysis: A Study in Econometrics|location=New York|publisher=John Wiley and Sons|series=Wiley publications in statistics |year=1953 |mr=64385}}
• {{Cite journal |last=Zhou |first=Lin |date=June 1993 |title=A simple proof of the Shapley–Folkman theorem |journal=Economic Theory |volume=3 |issue=2 |pages=371–372 |doi=10.1007/bf01212924 |issn=0938-2259}}

{{refend}}

• {{cite book|last=Starr|first=Ross M.|author-link=Ross Starr|chapter=8 Convex sets, separation theorems, and non-convex sets in {{math|R^N}} (Section 8.2.3 Measuring non-convexity, the Shapley–Folkman theorem)|title=General Equilibrium Theory: An Introduction|date=September 2009|pages=3–6|chapter-url=http://www.econ.ucsd.edu/~rstarr/113Winter2010/Webpage/PDFonlyCUPsubmission/Chap8-2009/2009CHAP-08092109.pdf|mr=1462618|id=(Draft of second edition, from Starr's course at the Economics Department of the University of California, San Diego)|doi=10.1017/CBO9781139174749|isbn=9781139174749|access-date=15 January 2011|archive-date=1 July 2010|archive-url=https://web.archive.org/web/20100701131416/http://www.econ.ucsd.edu/~rstarr/113Winter2010/Webpage/PDFonlyCUPsubmission/Chap8-2009/2009CHAP-08092109.pdf|url-status=dead}}
• {{cite web|last=Starr|first=Ross M.|author-link=Ross Starr|title=Shapley–Folkman theorem| date=May 2007 |pages=1–3|url=http://www.econ.ucsd.edu/~rstarr/SFarticle.pdf|access-date=15 January 2011|id=(Draft of article for the second edition of New Palgrave Dictionary of Economics)}}

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