nLab
convex set
Context
Topology
topology (point-set topology , point-free topology )

see also algebraic topology , functional analysis and topological homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents
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 .

One generalization of convexity to Riemannian manifold s and metric space s 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 October 1, 2015 21:54:33
by

Todd Trimble
(67.81.95.215)