nLab convex set




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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.


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

  • 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.

Last revised on November 15, 2023 at 04:21:56. See the history of this page for a list of all contributions to it.