The boundary of a convex set is always a convex curve. contain Euclidean space The intersection of all convex sets of a given subset A of A is called the convex hull of A. It is the smallest convex set containing A. Convex functions are real valued functions defined at intervals with attributes whose epigraph (set of points on or above the function graph) is a convex set. Convex minimization is a subfield of optimization. The minimization of convex functions on convex sets is studied. The branch of mathematics used to study the properties of convex sets and convex functions is called convex analysis. Let X be linear space 。 If for all x and y in subset S of X, and all t, points in interval [0,1] It also belongs to S, so S is called convex set 。 [4] The intersection of any convex set is a convex set. [5]
X's Subspace Is a convex set. If S is a convex set, x+S is also a convex set for any x in X. [4]
In other words, every point on the line connecting x and y is in C. This means practical or complex Topological linear space The convex set in is Road connectivity Of. In addition, if every point on the line segment connecting x and y except the endpoint is inside C, then C is strictly convex.
The convex subset (real number set) of R is only the interval of R. Some examples of convex subsets of Euclidean planes are solid regular polygons, intersections of solid triangles and solid triangles. Some examples of convex subsets of Euclidean three-dimensional space are Archimedean solids and Platonic solids. Kepler Bonoso polyhedron is an example of non convex set. [2]
A set that is not convex is called a nonconvex set. A polygon that is not a convex polygon is sometimes called a concave polygon. Some sources more generally use the term concave set to represent a non convex set, but most permissions prohibit this use.
The complement of a convex set is sometimes called an anticonvex set, especially in the context of mathematical optimization.
If S is a convex set in n-dimensional space, then for any r>1 in S, n-dimensional vector Set of, for any nonnegative number , that , then: This type of vector is called Of Convex combination 。 General:
order and Is a convex set, it has the following important properties: (1) Intersection Is a convex set. (2) And set Is a convex set. (3) Direct sum Is a convex set. The set of convex subsets of vector space has the following properties:
（1） empty set And the entire vector space is convex. (2) The convex point of any convex set set is convex.
(3) The union of a non decreasing sequence of a convex subset is a convex set. For the aforementioned attributes of the union of non decreasing sequences of convex sets, the restriction on nested sets is important: the union of two convex sets need not be convex.
A closed convex set is a convex set containing all its limit points. They can be represented as the intersection of a closed half space (a collection of points in space on one side of the hyperplane).
From what has just been said, it is obvious that such intersections are convex, and they are also closed. In order to prove the opposite, that is, each convex set can be expressed as such intersection, it is necessary to support the hyperplane theorem in the form of a given closed convex set C and its outer point P, there is a closed half space H, which contains C instead of P. The hyperplane theorem is the Hahn of functional analysis- Banach theorem Special circumstances. Each subset A of vector space is contained in the smallest convex set (called convex hull of A), that is, the intersection of all convex sets containing A. The convex hull operator Conv() has the characteristic attributes of the envelope set:
General: S ⊆ Conv (S);
Not decreasing S ⊆ T means that Conv (S) ⊆ Conv (T) and power are equal to Conv (Conv (S))=Conv (S).
Convex set operation is a set of convex sets required to form a lattice, where the "join" operation is the convex hull of the convex hull of two convex sets
Conv（S）∨Conv（T）= Conv（S∪T）= Conv（Conv（S）∪Conv（T））
The set of any set of convex sets is convex itself, so the convex subset of (real or compound) vector space forms a complete mesh.
In the actual vector space, the Minkowski sum of two (non empty) sets S1 and S2 is defined as the set S1+S2 formed by the vector set in the vector element set:
More generally, the Minkowski The sum is the vector passing through the element vector For Minkowski addition, the zero set {0} containing only zero vector 0 has special importance: for each non empty subset S of vector space
In algebraic terms, {0} is the ontological element of Minkowski addition (on the set of non empty sets).
Minkowski addition and convex hull
Minkowski addition performs well in the operation of obtaining convex hull, as shown in the following proposition:
Order S one ，S two Is a subset of the real vector space, which Minkowski sum The convex hull of For each finite set of non empty sets, this result is more general:
In mathematical terms, Minkowski's summation and convex hull forming operations are associated operations.
The sum of Minkowski's two compact convex sets is compact. The sum of compact convex sets and closed convex sets is closed. [3]
