Definition

A convex set is a subset of a real vector space with the property that, for any two points in the set, the straight line segment joining them lies entirely inside the set. In the familiar setting of Euclidean space, this means if x and y belong to a region S then every point of the form (1-t)x + t y with 0 <= t <= 1 also belongs to S. This elementary criterion captures the intuitive idea of a shape without indentations.

Key properties

  • Any convex combination (finite weighted average with nonnegative weights summing to one) of points in a convex set stays in the set.
  • Intersections of arbitrarily many convex sets are convex; unions need not be convex.
  • Affine images and preimages of convex sets are convex, so convexity is preserved under translation, scaling and linear maps.
  • Convex sets admit separation by hyperplanes under mild conditions, a fact used widely in analysis and optimization.

Examples and non-examples

Simple examples include intervals on the real line, disks and balls, convex polygons and polyhedra, and any affine subspace. The convex hull of a set is the smallest convex set containing it and is obtained by taking all convex combinations of points from the set. Typical non-examples are shapes with cavities or holes: a crescent, an annulus, or a star-shaped region that has inward dents are not convex because some segment between interior points leaves the region.

History and significance

The notion of convexity has roots in classical geometry and was developed further in the 19th and 20th centuries as part of convex geometry and functional analysis. Several deep results — for example, Helly-type intersection theorems, Carathéodory-type representation results and separation theorems — make convex sets central objects in both pure and applied mathematics.

Convex sets underlie convex optimization: when both the feasible region and the objective are convex, local minima are global minima, which simplifies analysis and computation. They also appear in economics (utility and production sets), probability (sets of distributions), and computational geometry (collision detection, convex hull algorithms). Related terms include strictly convex (no line segment lies on the boundary) and convex functions, which exhibit a convex epigraph — the set above the graph is a convex set. The boundary of a convex set is often formed by a convex curve or a union of flat faces in polyhedral cases.