Convex set#Non-convex set

File:Convex supergraph.svg is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.]]

Let {{mvar|S}} be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces). A subset {{mvar|C}} of {{mvar|S}} is convex if, for all {{mvar|x}} and {{mvar|y}} in {{mvar|C}}, the line segment connecting {{mvar|x}} and {{mvar|y}} is included in {{mvar|C}}.

This means that the affine combination {{math|(1 − t)x + ty}} belongs to {{mvar|C}} for all {{mvar|x,y}} in {{mvar|C}} and {{mvar|t}} in the interval {{math|[0, 1]}}. This implies that convexity is invariant under affine transformations. Further, it implies that a convex set in a real or complex topological vector space is path-connected (and therefore also connected).

A set {{mvar|C}} is {{visible anchor|strictly convex}} if every point on the line segment connecting {{mvar|x}} and {{mvar|y}} other than the endpoints is inside the topological interior of {{mvar|C}}. A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point.{{Halmos A Hilbert Space Problem Book 1982|p=5}}

A set {{mvar|C}} is absolutely convex if it is convex and balanced.

The convex subsets of {{math|R}} (the set of real numbers) are the intervals and the points of {{math|R}}. Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids. The Kepler-Poinsot polyhedra are examples of non-convex sets.

A set that is not convex is called a non-convex set. A polygon that is not a convex polygon is sometimes called a concave polygon,{{cite book |first=Jeffrey J. |last=McConnell |year=2006 |title=Computer Graphics: Theory Into Practice |isbn=0-7637-2250-2 |page=[https://archive.org/details/computergraphics0000mcco/page/130 130] |publisher=Jones & Bartlett Learning |url=https://archive.org/details/computergraphics0000mcco/page/130 }}. and some sources more generally use the term concave set to mean a non-convex set,{{MathWorld|title=Concave|id=Concave}} but most authorities prohibit this usage.{{cite book|title=Analytical Methods in Economics|first=Akira|last=Takayama|publisher=University of Michigan Press|year=1994|isbn=9780472081356|url=https://books.google.com/books?id=_WmZA0MPlmEC&pg=PA54|page=54|quote=An often seen confusion is a "concave set". Concave and convex functions designate certain classes of functions, not of sets, whereas a convex set designates a certain class of sets, and not a class of functions. A "concave set" confuses sets with functions.}}{{cite book|title=An Introduction to Mathematical Analysis for Economic Theory and Econometrics|first1=Dean|last1=Corbae|first2=Maxwell B.|last2=Stinchcombe|first3= Juraj|last3=Zeman|publisher=Princeton University Press|year=2009|isbn=9781400833085|url=https://books.google.com/books?id=j5P83LtzVO8C&pg=PT347|page=347|quote=There is no such thing as a concave set.}}

The complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization.{{cite journal | last = Meyer | first = Robert | journal = SIAM Journal on Control and Optimization | mr = 0312915 | pages = 41–54 | title = The validity of a family of optimization methods | volume = 8 | year = 1970| doi = 10.1137/0308003 | url = https://minds.wisconsin.edu/bitstream/handle/1793/57508/TR28.pdf?sequence=1 }}.

Given {{mvar|r}} points {{math|u₁, ..., u_r}} in a convex set {{mvar|S}}, and {{mvar|r}}

nonnegative numbers {{math|λ₁, ..., λ_r}} such that {{math|λ₁ + ... + λ_r {{=}} 1}}, the affine combination

$$\backslash sum\_\{k=1\}^r\backslash lambda\_k\; u\_k$$

belongs to {{mvar|S}}. As the definition of a convex set is the case {{math|1=r = 2}}, this property characterizes convex sets.

Such an affine combination is called a convex combination of {{math|u₁, ..., u_r}}. The convex hull of a subset {{mvar|S}} of a real vector space is defined as the intersection of all convex sets that contain {{mvar|S}}. More concretely, the convex hull is the set of all convex combinations of points in {{mvar|S}}. In particular, this is a convex set.

A (bounded) convex polytope is the convex hull of a finite subset of some Euclidean space {{math|Rⁿ}}.

The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties:Soltan, Valeriu, Introduction to the Axiomatic Theory of Convexity, Ştiinţa, Chişinău, 1984 (in Russian).

{{cite book|last=Singer|first=Ivan|title=Abstract convex analysis|series=Canadian Mathematical Society series of monographs and advanced texts|publisher=John Wiley & Sons, Inc.|location=New York|year= 1997|pages=xxii+491|isbn=0-471-16015-6|mr=1461544}}

1. The empty set and the whole space are convex.
2. The intersection of any collection of convex sets is convex.
3. The union of a collection of convex sets is convex if those sets form a chain (a totally ordered set) under inclusion. For this property, the restriction to chains is important, as the union of two convex sets need not be convex.

Closed convex sets are convex sets that contain all their limit points. They can be characterised as the intersections of closed half-spaces (sets of points in space that lie on and to one side of a hyperplane).

From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set {{mvar|C}} and point {{mvar|P}} outside it, there is a closed half-space {{mvar|H}} that contains {{mvar|C}} and not {{mvar|P}}. The supporting hyperplane theorem is a special case of the Hahn–Banach theorem of functional analysis.

A face of a convex set $C$ is a convex subset $F$ of $C$ such that whenever a point $p$ in $F$ lies strictly between two points $x$ and $y$ in $C$, both $x$ and $y$ must be in $F$.{{sfn | Rockafellar| 1997 | p=162}} Equivalently, for any $x,y\backslash in\; C$ and any real number trivial faces of $C$. An extreme point of $C$ is a point that is a face of $C$.

Let $C$ be a convex set in $\backslash R^n$ that is compact (or equivalently, closed and bounded). Then $C$ is the convex hull of its extreme points.{{sfn | Rockafellar| 1997 | p=166}} More generally, each compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points (the Krein–Milman theorem).

For example:

• A triangle in the plane (including the region inside) is a compact convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extreme points are the three vertices.)
• The only nontrivial faces of the closed unit disk $\backslash \{\; (x,y)\; \backslash in\; \backslash R^2:\; x^2+y^2\; \backslash leq\; 1\; \backslash \}$ are its extreme points, namely the points on the unit circle $S^1\; =\; \backslash \{\; (x,y)\; \backslash in\; \backslash R^2:\; x^2+y^2=1\; \backslash \}$.

Let {{mvar|C}} be a convex body in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle r in {{mvar|C}} such that a homothetic copy R of r is circumscribed about {{mvar|C}}. The positive homothety ratio is at most 2 and:{{Cite journal | doi = 10.1007/BF01263495| title = Approximation of convex bodies by rectangles| journal = Geometriae Dedicata| volume = 47| pages = 111–117| year = 1993| last1 = Lassak | first1 = M. | s2cid = 119508642}}

$$\backslash tfrac\{1\}\{2\}\; \backslash cdot\backslash operatorname\{Area\}(R)\; \backslash leq\; \backslash operatorname\{Area\}(C)\; \backslash leq\; 2\backslash cdot\; \backslash operatorname\{Area\}(r)$$

The set $\backslash mathcal\{K\}^2$ of all planar convex bodies can be parameterized in terms of the convex body diameter D, its inradius r (the biggest circle contained in the convex body) and its circumradius R (the smallest circle containing the convex body). In fact, this set can be described by the set of inequalities given by{{Cite journal|last=Santaló|first=L.|date=1961|title=Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa planas|journal=Mathematicae Notae|volume=17|pages=82–104}}{{Cite journal|last1=Brandenberg|first1=René|last2=González Merino|first2=Bernardo|date=2017|title=A complete 3-dimensional Blaschke-Santaló diagram|url=http://mia.ele-math.com/20-22|journal=Mathematical Inequalities & Applications|language=en|issue=2|pages=301–348|doi=10.7153/mia-20-22|issn=1331-4343|doi-access=free|arxiv=1404.6808}}

$$2r\; \backslash le\; D\; \backslash le\; 2R$$

$$R\; \backslash le\; \backslash frac\{\backslash sqrt\{3\}\}\{3\}\; D$$

$$r\; +\; R\; \backslash le\; D$$

$$D^2\; \backslash sqrt\{4R^2-D^2\}\; \backslash le\; 2R\; (2R\; +\; \backslash sqrt\{4R^2\; -D^2\})$$

and can be visualized as the image of the function g that maps a convex body to the {{math|R²}} point given by (r/R, D/2R). The image of this function is known a (r, D, R) Blachke-Santaló diagram.

File:Blaschke-Santaló_diagram_for_planar_convex_bodies.pdf and $\backslash mathbb\{B\}\_2$ the unit circle.]]

Alternatively, the set $\backslash mathcal\{K\}^2$ can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.

Let X be a topological vector space and $C\; \backslash subseteq\; X$ be convex.

• $\backslash operatorname\{Cl\}\; C$ and $\backslash operatorname\{Int\}\; C$ are both convex (i.e. the closure and interior of convex sets are convex).
• If $a\; \backslash in\; \backslash operatorname\{Int\}\; C$ and $b\; \backslash in\; \backslash operatorname\{Cl\}\; C$ then $[a,\; b[\; \backslash ,\; \backslash subseteq\; \backslash operatorname\{Int\}\; C$ (where ).
• If $\backslash operatorname\{Int\}\; C\; \backslash neq\; \backslash emptyset$ then:
• $\backslash operatorname\{cl\}\; \backslash left(\; \backslash operatorname\{Int\}\; C\; \backslash right)\; =\; \backslash operatorname\{Cl\}\; C$, and
• $\backslash operatorname\{Int\}\; C\; =\; \backslash operatorname\{Int\}\; \backslash left(\; \backslash operatorname\{Cl\}\; C\; \backslash right)\; =\; C^i$, where $C^\{i\}$ is the algebraic interior of C.

{{Main|convex hull}}

Every subset {{mvar|A}} of the vector space is contained within a smallest convex set (called the convex hull of {{mvar|A}}), namely the intersection of all convex sets containing {{mvar|A}}. The convex-hull operator Conv() has the characteristic properties of a closure operator:

• extensive: {{math|S ⊆ Conv(S)}},
• non-decreasing: {{math|S ⊆ T}} implies that {{math|Conv(S) ⊆ Conv(T)}}, and
• idempotent: {{math|Conv(Conv(S)) {{=}} Conv(S)}}.

The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets

$$\backslash operatorname\{Conv\}(S)\backslash vee\backslash operatorname\{Conv\}(T)\; =\; \backslash operatorname\{Conv\}(S\backslash cup\; T)\; =\; \backslash operatorname\{Conv\}\backslash bigl(\backslash operatorname\{Conv\}(S)\backslash cup\backslash operatorname\{Conv\}(T)\backslash bigr).$$

The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice.

{{Main|Minkowski addition}}

File:Minkowski sum graph - vector version.svg of sets. The sum of the squares Q₁=[0,1]² and Q₂=[1,2]² is the square Q₁+Q₂=[1,3]².]]

In a real vector-space, the Minkowski sum of two (non-empty) sets, {{math|S₁}} and {{math|S₂}}, is defined to be the set {{math|S₁ + S₂}} formed by the addition of vectors element-wise from the summand-sets

$$S\_1+S\_2=\backslash \{x\_1+x\_2:\; x\_1\backslash in\; S\_1,\; x\_2\backslash in\; S\_2\backslash \}.$$

More generally, the Minkowski sum of a finite family of (non-empty) sets {{math|S_n}} is the set formed by element-wise addition of vectors

$$\backslash sum\_n\; S\_n\; =\; \backslash left\; \backslash \{\; \backslash sum\_n\; x\_n\; :\; x\_n\; \backslash in\; S\_n\; \backslash right\; \backslash \}.$$

For Minkowski addition, the zero set {{math|{0} }} containing only the zero vector {{math|0}} has special importance: For every non-empty subset S of a vector space

$$S+\backslash \{0\backslash \}=S;$$

in algebraic terminology, {{math|{0} }} is the identity element of Minkowski addition (on the collection of non-empty sets).The empty set is important in Minkowski addition, because the empty set annihilates every other subset: For every subset {{mvar|S}} of a vector space, its sum with the empty set is empty: $S+\backslash emptyset=\backslash emptyset$.

Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:

Let {{math|S₁, S₂}} be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls

$$\backslash operatorname\{Conv\}(S\_1+S\_2)=\backslash operatorname\{Conv\}(S\_1)+\backslash operatorname\{Conv\}(S\_2).$$

This result holds more generally for each finite collection of non-empty sets:

$$\backslash text\{Conv\}\backslash left\; (\; \backslash sum\_n\; S\_n\; \backslash right\; )\; =\; \backslash sum\_n\; \backslash text\{Conv\}\; \backslash left\; (S\_n\; \backslash right).$$

In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.Theorem 3 (pages 562–563): {{cite journal|first1=M.|last1=Krein|author-link1=Mark Krein|first2=V.|last2=Šmulian|year=1940|title=On regularly convex sets in the space conjugate to a Banach space|journal=Annals of Mathematics |series=Second Series| volume=41 |issue=3 |pages=556–583|jstor=1968735|doi=10.2307/1968735}}For the commutativity of Minkowski addition and convexification, see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the convex hulls of Minkowski sumsets in its "Chapter 3 Minkowski addition" (pages 126–196): {{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=0-521-35220-7|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}

The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed.Lemma 5.3: {{cite book|first1=C.D.|last1= Aliprantis|first2=K.C.| last2=Border|title=Infinite Dimensional Analysis, A Hitchhiker's Guide| publisher=Springer| location=Berlin|year=2006|isbn=978-3-540-29587-7}}

The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed.{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|url=https://archive.org/details/convexanalysisge00zali_934|url-access=limited|publisher=World Scientific Publishing Co., Inc|location= River Edge, NJ |date= 2002|page=[https://archive.org/details/convexanalysisge00zali_934/page/n27 7]|isbn=981-238-067-1|mr=1921556}} It uses the concept of a recession cone of a non-empty convex subset S, defined as:

$$\backslash operatorname\{rec\}\; S\; =\; \backslash left\backslash \{\; x\; \backslash in\; X\; \backslash ,\; :\; \backslash ,\; x\; +\; S\; \backslash subseteq\; S\; \backslash right\backslash \},$$

where this set is a convex cone containing $0\; \backslash in\; X$ and satisfying $S\; +\; \backslash operatorname\{rec\}\; S\; =\; S$. Note that if S is closed and convex then $\backslash operatorname\{rec\}\; S$ is closed and for all $s\_0\; \backslash in\; S$,

$$\backslash operatorname\{rec\}\; S\; =\; \backslash bigcap\_\{t\; >\; 0\}\; t\; (S\; -\; s\_0).$$

Theorem (Dieudonné). Let A and B be non-empty, closed, and convex subsets of a locally convex topological vector space such that $\backslash operatorname\{rec\}\; A\; \backslash cap\; \backslash operatorname\{rec\}\; B$ is a linear subspace. If A or B is locally compact then A − B is closed.

The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name "generalized convexity" is used, because the resulting objects retain certain properties of convex sets.

{{main|Star domain}}

Let {{mvar|C}} be a set in a real or complex vector space. {{mvar|C}} is star convex (star-shaped) if there exists an {{math|x₀}} in {{mvar|C}} such that the line segment from {{math|x₀}} to any point {{mvar|y}} in {{mvar|C}} is contained in {{mvar|C}}. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.

{{main|Orthogonal convex hull}}

An example of generalized convexity is orthogonal convexity.Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations", in: Computational Morphology, 137-152. Elsevier, 1988.

A set {{mvar|S}} in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of {{mvar|S}} lies totally within {{mvar|S}}. It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. Some other properties of convex sets are valid as well.

The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set.

Convexity can be extended for a totally ordered set {{mvar|X}} endowed with the order topology.Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). {{ISBN|0-13-181629-2}}.

Let {{math|Y ⊆ X}}. The subspace {{mvar|Y}} is a convex set if for each pair of points {{math|a, b}} in {{mvar|Y}} such that {{math|a ≤ b}}, the interval {{math|[a, b] {{=}} {x ∈ X {{!}} a ≤ x ≤ b} }} is contained in {{mvar|Y}}. That is, {{mvar|Y}} is convex if and only if for all {{math|a, b}} in {{mvar|Y}}, {{math|a ≤ b}} implies {{math|[a, b] ⊆ Y}}.

A convex set is {{em|not}} connected in general: a counter-example is given by the subspace {1,2,3} in {{math|Z}}, which is both convex and not connected.

The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms.

Given a set {{mvar|X}}, a convexity over {{mvar|X}} is a collection {{math|𝒞}} of subsets of {{mvar|X}} satisfying the following axioms:{{cite book|last=van De Vel|first=Marcel L. J.|title=Theory of convex structures|series=North-Holland Mathematical Library|publisher=North-Holland Publishing Co.|location=Amsterdam|year= 1993|pages=xvi+540|isbn=0-444-81505-8|mr=1234493}}

1. The empty set and {{mvar|X}} are in {{math|𝒞}}.
2. The intersection of any collection from {{math|𝒞}} is in {{math|𝒞}}.
3. The union of a chain (with respect to the inclusion relation) of elements of {{math|𝒞}} is in {{math|𝒞}}.

The elements of {{math|𝒞}} are called convex sets and the pair {{math|(X, 𝒞)}} is called a convexity space. For the ordinary convexity, the first two axioms hold, and the third one is trivial.

For an alternative definition of abstract convexity, more suited to discrete geometry, see the convex geometries associated with antimatroids.

{{main|Convex space}}

Convexity can be generalised as an abstract algebraic structure: a space is convex if it is possible to take convex combinations of points.

{{Div col|colwidth=25em}}

• Absorbing set
• Algorithmic problems on convex sets
• Bounded set (topological vector space)
• Brouwer fixed-point theorem
• Complex convexity
• Convex cone
• Convex series
• Convex metric space
• Carathéodory's theorem (convex hull)
• Choquet theory
• Helly's theorem
• Holomorphically convex hull
• Integrally-convex set
• John ellipsoid
• Pseudoconvexity
• Radon's theorem
• Shapley–Folkman lemma
• Symmetric set

{{Div col end}}

{{reflist|30em}}

• {{cite book | last=Rockafellar | first=R. T. | author-link=R. Tyrrell Rockafellar | title=Convex Analysis |publisher=Princeton University Press | location=Princeton, NJ | orig-year=1970 | year=1997 | isbn=1-4008-7317-7 |url=https://books.google.com/books?id=1TiOka9bx3sC }}

{{Wiktionary}}

• {{springer|title=Convex subset|id=p/c026380|mode=cs1}}
• [http://www.fmf.uni-lj.si/~lavric/lauritzen.pdf Lectures on Convex Sets], notes by Niels Lauritzen, at Aarhus University, March 2010.

{{Functional analysis}}

{{Convex analysis and variational analysis}}

{{Authority control}}

{{DEFAULTSORT:Convex Set}}

Category:Convex analysis

Category:Convex geometry