nLab
convex set

Contents

Definition

A subset SS of a real affine space XX is convex if for any two points x,ySx,y \in S, the straight line segment connecting xx with yy in XX is also contained in SS. In other words, for any x,ySx,y\in S, and any t[0,1]t\in [0,1], we have also tx+(1t)ySt x + (1-t) y \in S.

Properties

Every convex set is star-shaped.

  • One generalization of convexity to Riemannian manifolds and metric spaces is geodesic convexity.

  • An abstract generalization of the notion of a convex set is that of a convex space. Note that as mentioned there, there is a nice characterization of those convex spaces which are isomorphic to convex subsets of real affine space.

  • The convex hull of a subset is the smallest convex subset containing it.

Revised on August 27, 2013 07:28:07 by Colin Tan (183.90.103.174)