Convexity in economics

{{off topic|convex analysis|date=August 2013}}

The economics depends upon the following definitions and results from convex geometry.

{{multiple image

| width = 107

| footer = Line segments test convexity.

| 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 lies in the interior of the convex set.

| caption1 = A convex set covers the line segment connecting any two of its points.

| image2 = Convex polygon illustration2.svg

| alt2 = Illustration of a non‑convex set, which looks somewhat like a boomerang or cashew nut. A (green) non‑convex set contains the (black) line segment joining the points x and y. Part of the line segment lies outside of the (green) non‑convex set.

| caption2 = A non‑convex set fails to cover a point in some line segment joining two of its points.

}}

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

: (x₁, y₁) + (x₂, y₂) = (x₁+x₂, y₁+y₂);

further, a point can be multiplied by each real number λ coordinate-wise

: λ (x, y) = (λx, λy).

More generally, any real vector space of (finite) dimension D can be viewed as the set of all possible lists of D real numbers {{nowrap|{ (v₁, v₂, . . . , v_D)}} } together with two operations: 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.

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

In a real vector space, a set is defined to be convex if, for each pair of its points, every point on the line segment that joins them is covered by the set. For example, a solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non‑convex. Trivially, the empty set is convex.

More formally, a set Q is convex if, for all points v₀ and v₁ in Q and for every real number λ in the unit interval {{closed-closed|0,1}}, the point

: (1 − λ) v₀ + λv₁

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 an indexed subset {v₀, v₁, . . . , v_D} of a vector space is any weighted average {{nowrap|λ₀v₀ + λ₁v₁ + . . . + λ_Dv_D,}} for some indexed set of non‑negative real numbers {λ_d} satisfying the equation {{nowrap|λ₀ + λ₁ + . . . + λ_D}} = 1.

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.

For every subset Q of a real vector space, its {{nowrap|convex hull Conv(Q)}} is the minimal convex set that contains Q. Thus Conv(Q) is the intersection of all the convex sets that cover Q. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in Q.

File:Supporting hyperplane1.svg $S$ (in pink), a supporting hyperplane of $S$ (the dashed line), and the half-space delimited by the hyperplane that contains $S$ (in light blue). ]]

Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set $S$ in the real n-space $\backslash mathbb\; R^n$ if it meets both of the following:

• $S$ is entirely contained in one of the two closed half-spaces determined by the hyperplane
• $S$ has at least one point on the hyperplane.

Here, a closed half-space is the half-space that includes the hyperplane.

File:Supporting hyperplane2.svg

This theorem states that if $S$ is a closed convex set in $\backslash mathbb\; R^n,$ and $x$ is a point on the boundary of $S,$ then there exists a supporting hyperplane containing $x.$

The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set $S$ is not convex, the statement of the theorem is not true at all points on the boundary of $S,$ as illustrated in the third picture on the right.

File:Supporting hyperplane3.svg

File:Indifference curves showing budget line.svg I₂.]]

An optimal basket of goods occurs where the consumer's convex preference set is supported by the budget constraint, as shown in the diagram. If the preference set is convex, then the consumer's set of optimal decisions is a convex set, for example, a unique optimal basket (or even a line segment of optimal baskets).

For simplicity, we shall assume that the preferences of a consumer can be described by a utility function that is a continuous function, which implies that the preference sets are closed. (The meanings of "closed set" is explained below, in the subsection on optimization applications.)

{{Main|Non-convexity (economics)}}

{{See also|Shapley–Folkman lemma}}

File:NonConvex.gif

If a preference set is non‑convex, then some prices produce a budget supporting two different optimal consumption decisions. 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 a half an eagle and a {{nowrap|half a lion}} (or a griffin)! Thus, the contemporary zoo-keeper's preferences are non‑convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.

Non‑convex sets have been incorporated in the theories of general economic equilibria,Pages 392–399 and page 188: {{cite book |last1=Arrow |first1=Kenneth J. |author-link1=Kenneth Arrow |last2=Hahn |first2=Frank H. |author-link2=Frank Hahn |year=1971 |chapter=Appendix B: Convex and related sets |title=General competitive analysis |chapter-url=https://archive.org/details/generalcompetiti0000arro |chapter-url-access=registration |publisher=Holden-Day, Inc. [North-Holland] |pages=[https://archive.org/details/generalcompetiti0000arro/page/375 375–401] |mr=439057 |series=Mathematical economics texts [Advanced textbooks in economics] |number=6 [12] |location=San Francisco |isbn=978-0-444-85497-1}}
Pages 52–55 with applications on pages 145–146, 152–153, and 274–275: {{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 |number=9 |publisher=Cambridge University Press |isbn=978-0-521-26514-0 |mr=1113262}}
Theorem C(6) on page 37 and applications on pages 115–116, 122, and 168: {{cite book |last=Hildenbrand |first=Werner |author-link=Werner Hildenbrand |title=Core and equilibria of a large economy |series=Princeton studies in mathematical economics |number=5 |publisher=Princeton University Press |year=1974 |pages=viii+251 |isbn=978-0-691-04189-6 |mr=389160}} of market failures,Pages 112–113 in Section 7.2 "Convexification by numbers" (and more generally pp. 107–115): {{cite book |last=Salanié |first=Bernard |chapter=7 Nonconvexities |title=Microeconomics of market failures |edition=English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris) |year=2000 |publisher=MIT Press |pages=107–125 |isbn=978-0-262-19443-3}} and of public economics.Pages 63–65: {{cite book |last=Laffont |first=Jean-Jacques |author-link=Jean-Jacques Laffont |year=1988 |chapter=3 Nonconvexities |title=Fundamentals of public economics |chapter-url=https://books.google.com/books?id=O5MnAQAAIAAJ&q=editions:ISBN%200-262-12127-1 |publisher=MIT |isbn=978-0-262-12127-9 |url-access=registration |url=https://archive.org/details/fundamentalsofpu0000laff }} These results are described in graduate-level textbooks in microeconomics,{{cite book |author-link=Hal Varian |last=Varian |first=Hal R. |chapter=21.2 Convexity and size |pages=[https://archive.org/details/microeconomicana00vari_0/page/393 393–394] |title=Microeconomic Analysis |publisher=W. W. Norton & Company |edition=3rd |year=1992 |isbn=978-0-393-95735-8 |mr=1036734 |chapter-url-access=registration |chapter-url=https://archive.org/details/microeconomicana00vari_0 |url=https://archive.org/details/microeconomicana00vari_0/page/393 }}
Page 628: {{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 |pages=627–630 |isbn=978-0-19-507340-9}} general equilibrium theory,Page 169 in the first edition: {{cite book |last=Starr |first=Ross M. |chapter=8 Convex sets, separation theorems, and non‑convex sets in R^N |title=General equilibrium theory: An introduction |edition=Second |publisher=Cambridge University Press |location=Cambridge |year=2011 |isbn=978-0-521-53386-7 |mr=1462618|doi=10.1017/CBO9781139174749}}
In Ellickson, page xviii, and especially Chapter 7 "Walras meets Nash" (especially section 7.4 "Nonconvexity" pages 306–310 and 312, and also 328–329) and Chapter 8 "What is Competition?" (pages 347 and 352): {{cite book |title=Competitive equilibrium: Theory and applications |first=Bryan |last=Ellickson |publisher=Cambridge University Press |isbn=978-0-521-31988-1 |year=1994 |pages=420}}

game theory,Theorem 1.6.5 on pages 24–25: {{cite book |last=Ichiishi |first=Tatsuro |title=Game theory for economic analysis |series=Economic theory, econometrics, and mathematical economics |publisher=Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers] |location=New York |year=1983 |pages=x+164 |isbn=978-0-12-370180-0 |mr=700688}} mathematical economics,Pages 127 and 33–34: {{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, New York |year=1981 |pages=xi+145 |isbn=978-0-521-28614-5 |mr=657578}}

and applied mathematics (for economists).Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: {{cite book |last=Carter |first=Michael |title=Foundations of mathematical economics |publisher=MIT Press |year=2001 |pages=xx+649 |isbn=978-0-262-53192-4 |mr=1865841}}
Page 309: {{cite book |last=Moore |first=James C. |title=Mathematical methods for economic theory: Volume I

|series=Studies in economic theory |volume=9 |publisher=Springer-Verlag |location=Berlin |year=1999 |pages=xii+414 |isbn=978-3-540-66235-8 |mr=1727000|doi=10.1007/978-3-662-08544-8 }}
Pages 47–48: {{cite book |mr=1878374 |last1=Florenzano |first1=Monique |last2=Le Van |first2=Cuong |title=Finite dimensional convexity and optimization |others=in cooperation with Pascal Gourdel |s2cid=117240618 |series=Studies in economic theory |volume=13 |publisher=Springer-Verlag |location=Berlin |year=2001 |pages=xii+154 |isbn=978-3-540-41516-9|doi=10.1007/978-3-642-56522-9}} The Shapley–Folkman lemma results establish that non‑convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to production economies with many small firms.Economists have studied non‑convex sets using advanced mathematics, particularly differential geometry and topology, Baire category, measure and integration theory, and ergodic theory: {{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 |pages=viii+205 |isbn=978-3-540-12881-6 |mr=737006|doi=10.1007/978-3-642-46488-1 }}

In "oligopolies" (markets dominated by a few producers), especially in "monopolies" (markets dominated by one producer), non‑convexities remain important.Page 1: {{cite journal |last=Guesnerie |first=Roger |author-link=Roger Guesnerie |title=Pareto optimality in non‑convex economies |journal=Econometrica |volume=43 |issue=1 |year=1975 |pages=1–29 |jstor=1913410 |doi=10.2307/1913410 |mr=443877}} ({{cite journal |title=Errata |journal=Econometrica |volume=43 |year=1975 |number=5–6 |page=1010 |jstor=1911353 |doi=10.2307/1911353 |mr=443878|last1=Guesnerie |first1=Roger}}) Concerns with large producers exploiting market power in fact initiated the literature on non‑convex sets, when Piero Sraffa wrote about on firms with increasing returns to scale in 1926,{{cite journal |last=Sraffa |first=Piero |s2cid=6458099 |author-link=Piero Sraffa |year=1926 |title=The Laws of returns under competitive conditions |journal=Economic Journal |volume=36 |number=144 |pages=535–550 |jstor=2959866 |doi=10.2307/2959866}} after which Harold Hotelling wrote about marginal cost pricing in 1938.{{cite journal |

first=Harold |last=Hotelling |author-link=Harold Hotelling |title=The General welfare in relation to problems of taxation and of railway and utility rates |journal=Econometrica |volume=6 |number=3 |date=July 1938 |pages=242–269 |jstor=1907054 |doi=10.2307/1907054}} Both Sraffa and Hotelling illuminated the market power of producers without competitors, clearly stimulating a literature on the supply-side of the economy.Pages 5–7: {{cite book |last=Quinzii |first=Martine|author-link= Martine Quinzii |title=Increasing returns and efficiency |location=New York |publisher=Oxford University Press |year=1992 |edition=Revised translation of (1988) Rendements croissants et efficacité economique. Paris: Editions du Centre National de la Recherche Scientifique |pages=viii+165 |isbn=978-0-19-506553-4}}

Non‑convex sets arise also with environmental goods (and other externalities),Pages 106, 110–137, 172, and 248: {{cite book |title=The Theory of environmental policy |edition=Second |first1=William J. |last1=Baumol |author-link1=William Baumol |last2=Oates |first2=Wallace E. |isbn=978-0-521-31112-0 |year=1988 |pages=x+299 |publisher=Cambridge University Press |location=Cambridge |others=with contributions by V. S. Bawa and David F. Bradford |chapter=8 Detrimental externalities and nonconvexities in the production set}}

{{cite journal |mr=449575 |last=Starrett |first=David A. |title=Fundamental nonconvexities in the theory of externalities |journal=Journal of Economic Theory |volume=4 |year=1972 |number=2 |pages=180–199 |doi=10.1016/0022-0531(72)90148-2 }}
Starrett discusses non‑convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): {{cite book |last=Starrett |first=David A. |title=Foundations of public economics |series=Cambridge economic handbooks |year=1988 |publisher=Cambridge University Press |location=Cambridge |url=https://archive.org/details/foundationsofpub0000star |url-access=registration |quote=nonconvex OR nonconvexities. |isbn=9780521348010 }} with information economics,{{cite journal |first=Roy |last=Radner |author-link=Roy Radner |title=Competitive equilibrium under uncertainty |journal=Econometrica |volume=36 |issue=1 |year=1968 |pages=31–53 |doi=10.2307/1909602|jstor=1909602 }}

and with stock markets (and other incomplete markets).Page 270: {{cite book |mr=926685 |last=Drèze |first=Jacques H. |author-link=Jacques H. Drèze |title=Essays on economic decisions under uncertainty |publisher=Cambridge University Press |editor-last=Drèze |editor-first=J. H. |location=Cambridge |year=1987 | pages=261–297 |isbn=978-0-521-26484-6 |chapter=14 Investment under private ownership: Optimality, equilibrium and stability |doi=10.1017/CBO9780511559464 }} (Originally published as {{cite book |last=Drèze |first=Jacques H. |author-link=Jacques H. Drèze |year=1974 |chapter=Investment under private ownership: Optimality, equilibrium and stability |editor-last=Drèze |editor-first=J. H.|title=Allocation under Uncertainty: Equilibrium and Optimality |publisher=Wiley |location=New York |pages=129–165 }})Page 371: {{cite book |last1=Magill |first1=Michael |last2=Quinzii |first2=Martine |author2-link= Martine Quinzii |year=1996 |chapter=6 Production in a finance economy, Section 31 Partnerships|pages=329–425 |title=The Theory of incomplete markets |publisher=MIT Press |location=Cambridge, Massachusetts}} Such applications continued to motivate economists to study non‑convex sets.{{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 |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |publisher=Palgrave Macmillan |year=1987 |edition=first|doi=10.1057/9780230226203.3173 |pages=653–661 |chapter-url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf |isbn=9780333786765}}

{{cleanup|section|date=August 2013|reason=Relationship between subderivatives and non‑convexity remains cryptic}}

Economists have increasingly studied non‑convex sets with nonsmooth analysis, which generalizes convex analysis. "Non‑convexities in [both] production and consumption ... required mathematical tools that went beyond convexity, and further development had to await the invention of non‑smooth calculus" (for example, Francis Clarke's locally Lipschitz calculus), as described by {{harvtxt|Rockafellar|Wets|1998}}{{cite book |last1=Rockafellar |first1=R. Tyrrell |author-link1=R. Tyrrell Rockafellar |last2=Wets |first2=Roger J-B |s2cid=198120391 |author-link2=Roger J-B Wets |title=Variational analysis |series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] |volume=317 |publisher=Springer-Verlag |location=Berlin |year=1998 |pages=xiv+733 |isbn=978-3-540-62772-2 |mr=1491362 |doi=10.1007/978-3-642-02431-3 }} and {{harvtxt|Mordukhovich|2006}},Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495:
{{cite book |author-link=Boris Mordukhovich |first=Boris S. |last=Mordukhovich |title=Variational analysis and generalized differentiation II: Applications |series=Grundlehren Series (Fundamental Principles of Mathematical Sciences) |volume=331 |publisher=Springer |year=2006 |pages=i–xxii and 1–610 |mr=2191745 }} according to {{harvtxt|Khan|2008}}.{{cite book |last=Khan |first=M. Ali |chapter=Perfect competition |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E. |editor2-last=Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= 354–365|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_P000056 |doi=10.1057/9780230226203.1267 |isbn=978-0-333-78676-5}} {{harvtxt|Brown|1991|pp=1967–1968}} wrote that the "major methodological innovation in the general equilibrium analysis of firms with pricing rules" was "the introduction of the methods of non‑smooth analysis, as a [synthesis] of global analysis (differential topology) and [of] convex analysis." According to {{harvtxt|Brown|1991|p=1966}}, "Non‑smooth analysis extends the local approximation of manifolds by tangent planes [and extends] the analogous approximation of convex sets by tangent cones to sets" that can be non‑smooth or non‑convex.{{cite book |last=Brown |first=Donald J. |chapter=36 Equilibrium analysis with non‑convex technologies |doi=10.1016/S1573-4382(05)80011-6 |title=Handbook of mathematical economics, Volume IV |pages=1963–1995 [1966] |mr=1207195 |editor1-first=Werner |editor1-last=Hildenbrand |editor1-link=Werner Hildenbrand |editor2-first=Hugo |editor2-last=Sonnenschein |editor2-link=Hugo F. Sonnenschein |series=Handbooks in Economics |volume=1 |publisher=North-Holland Publishing Co |location=Amsterdam |year=1991 |isbn=0-444-87461-5 }} Economists have also used algebraic topology.{{cite journal |mr=1218037 |last=Chichilnisky |first=G. |author-link=Graciela Chichilnisky |title=Intersecting families of sets and the topology of cones in economics |journal=Bulletin of the American Mathematical Society |series=New Series |volume=29 |year=1993 |number=2 |pages=189–207 |doi=10.1090/S0273-0979-1993-00439-7 |url=https://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00439-7/S0273-0979-1993-00439-7.pdf |arxiv=math/9310228 |citeseerx=10.1.1.234.3909 |bibcode=1993math.....10228C}}

• Convex duality

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{{reflist}}

• {{cite book |last=Blume |first=Lawrence E. |author-link=Lawrence E. Blume |chapter=Convexity |year=2008a |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= 225–226|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508 |doi=10.1057/9780230226203.0315 |isbn=978-0-333-78676-5}}
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• {{cite book |last=Blume |first=Lawrence E. |chapter=Duality |year=2008c |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= 551–555|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |doi=10.1057/9780230226203.0411 |isbn=978-0-333-78676-5}}
• {{cite book |last=Crouzeix |first=J.-P. |chapter=Quasi-concavity |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E |editor2-last=Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= 815–816|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008 |doi=10.1057/9780230226203.1375 |isbn=978-0-333-78676-5}}
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• {{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 |editor2-first=Murray |editor2-last=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.econ.upf.edu/~mcolell/research/art_083b.pdf |isbn=9780333786765}}
• {{cite book |last=Newman |first=Peter |author-link=Peter Kenneth Newman |chapter=Convexity |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987c |edition=first|doi=10.1057/9780230226203.2282 |pages= 1|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000453 |isbn=9780333786765}}
• {{cite book |last=Newman |first=Peter |author-link=Peter Kenneth Newman |chapter=Duality |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987d |edition=first|doi=10.1057/9780230226203.2412 |pages= 1|chapter-url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |isbn=9780333786765}}
• {{cite book |last=Rockafellar |first=R. Tyrrell |author-link=R. Tyrrell Rockafellar|title=Convex analysis |edition=Reprint of the 1979 Princeton mathematical series 28 |series=Princeton landmarks in mathematics |publisher=Princeton University Press |location=Princeton, NJ |year=1997|isbn=978-0-691-01586-6|mr=274683}}.
• {{cite book |last=Schneider |first=Rolf |title=Convex bodies: The Brunn–Minkowski theory |series=Encyclopedia of mathematics and its applications |volume=44 |publisher=Cambridge University Press |location=Cambridge |year=1993 |pages=xiv+490 |isbn=978-0-521-35220-8 |mr=1216521 |doi=10.1017/CBO9780511526282 |url=https://archive.org/details/convexbodiesbrun0000schn}}

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Category:Convex hulls

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