Convex set

The convex subsets of R (the set of real numbers) are the intervals and the points of 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,[4] and some sources more generally use the term concave set to mean a non-convex set,[5] but most authorities prohibit this usage.[6][7]

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.[8]

The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties:[9][10]

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 sequence of convex sets is convex, if they form a non-decreasing chain for 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 point 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 C and point P outside it, there is a closed half-space H that contains C and not P. The supporting hyperplane theorem is a special case of the Hahn–Banach theorem of functional analysis.

Let C be a convex body in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C. The positive homothety ratio is at most 2 and:[11]

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

The set ${\displaystyle {\mathcal {K}}^{2}}$ 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[12][13]

$${\displaystyle 2r\leq D\leq 2R}$$

$${\displaystyle R\leq {\frac {\sqrt {3}}{3}}D}$$

$${\displaystyle r+R\leq D}$$

$${\displaystyle D^{2}{\sqrt {4R^{2}-D^{2}}}\leq 2R(2R+{\sqrt {4R^{2}-D^{2}}})}$$

and can be visualized as the image of the function g that maps a convex body to the R² point given by (r/R, D/2R). The image of this function is known a (r, D, R) Blachke-Santaló diagram.[13]

Blaschke-Santaló (r, D, R) diagram for planar convex bodies. ${\displaystyle \mathbb {L} }$ denotes the line segment, ${\displaystyle \mathbb {I} _{\frac {\pi }{3}}}$ the equilateral triangle, ${\displaystyle \mathbb {RT} }$ the Reuleaux triangle and ${\displaystyle \mathbb {B} _{2}}$ the unit circle.

Alternatively, the set ${\displaystyle {\mathcal {K}}^{2}}$ can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.[12][13]

Let X be a topological vector space and ${\displaystyle C\subseteq X}$ be convex.

• ${\displaystyle \operatorname {Cl} C}$ and ${\displaystyle \operatorname {Int} C}$ are both convex (i.e. the closure and interior of convex sets are convex).
• If ${\displaystyle a\in \operatorname {Int} C}$ and ${\displaystyle b\in \operatorname {Cl} C}$ then ${\displaystyle [a,b[\,\subseteq \operatorname {Int} C}$ (where ${\displaystyle [a,b[\,:=\left\{(1-r)a+rb:0\leq r<1\right\}}$).
• If ${\displaystyle \operatorname {Int} C\neq \emptyset }$ then:
  • ${\displaystyle \operatorname {cl} \left(\operatorname {Int} C\right)=\operatorname {Cl} C}$, and
  • ${\displaystyle \operatorname {Int} C=\operatorname {Int} \left(\operatorname {Cl} C\right)=C^{i}}$, where ${\displaystyle C^{i}}$ is the algebraic interior of C.

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

• extensive: S ⊆ Conv(S),
• non-decreasing: S ⊆ T implies that Conv(S) ⊆ Conv(T), and
• idempotent: 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

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

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.

Minkowski addition 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, S₁ and S₂, is defined to be the set S₁ + S₂ formed by the addition of vectors element-wise from the summand-sets

$${\displaystyle S_{1}+S_{2}=\{x_{1}+x_{2}:x_{1}\in S_{1},x_{2}\in S_{2}\}.}$$

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

$${\displaystyle \sum _{n}S_{n}=\left\{\sum _{n}x_{n}:x_{n}\in S_{n}\right\}.}$$

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

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

in algebraic terminology, {0} is the identity element of Minkowski addition (on the collection of non-empty sets).[14]

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

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

$${\displaystyle \operatorname {Conv} (S_{1}+S_{2})=\operatorname {Conv} (S_{1})+\operatorname {Conv} (S_{2}).}$$

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

$${\displaystyle {\text{Conv}}\left(\sum _{n}S_{n}\right)=\sum _{n}{\text{Conv}}\left(S_{n}\right).}$$

In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.[15][16]

The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed.[17]

The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed.[18] It uses the concept of a recession cone of a non-empty convex subset S, defined as:

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

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

$${\displaystyle \operatorname {rec} S=\bigcap _{t>0}t(S-s_{0}).}$$

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

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

An example of generalized convexity is orthogonal convexity.[19]

A set 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 S lies totally within 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 X endowed with the order topology.[20]

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

A convex set is not connected in general: a counter-example is given by the subspace {1,2,3} in 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 X, a convexity over X is a collection 𝒞 of subsets of X satisfying the following axioms:[9][10][21]

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

The elements of 𝒞 are called convex sets and the pair (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.

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