nLab convex set

Definition

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

Properties

Every convex set is star-shaped about each of its points, and hence contractible provided it is inhabited.

Related notions

• 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. 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.