nLab
convex set

Definition

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

Properties

Every convex set is star-shaped.

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. 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.