convexes

Convexes, in geometric and analytical contexts, are convex sets in a vector space. A set C is convex if, for any two points x and y in C, the line segment joining x and y lies entirely in C. Equivalently, C contains every convex combination (1−t)x + t y with 0 ≤ t ≤ 1 of points in C. This property makes many problems tractable, since mixtures of feasible points remain feasible.

Key properties and constructions preserve convexity. The intersection of any collection of convex sets is convex, and the convex hull of a set is the smallest convex set containing it. Affine transformations map convex sets to convex sets, and the Minkowski sum of two convex sets is convex.

Within a convex set, certain points called extreme points cannot be written as nontrivial convex combinations of other points in the set. Every point in a compact convex set is a convex combination of extreme points (Krein–Milman-type considerations apply in the appropriate setting). Hyperplanes can support convex sets at boundary points, and separation theorems guarantee the existence of separating hyperplanes between disjoint convex sets.

Common examples include convex polytopes, such as triangles and cubes, as well as smooth shapes like discs and ellipsoids. Nonconvex shapes fail the defining property; for instance, a crescent contains two points whose connecting segment leaves the set.

Convexes play a central role in convex analysis and optimization. Many optimization problems become easier when the feasible region is convex, leading to the study of convex functions, convex programs, and duality. Related concepts include convex hulls, polytopes, and support functions.