convex set

Let S a subset of n. We say that S is convex when, for any pair of points A,B in S, the segment AB¯ lies entirely inside S.

The former statement is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that for any pair of vectors u,v in S, the vector (1-t)u+tv is in S for all t[0,1].

If S is a convex set, for any u1,u2,,ur in S, and any positive numbers λ1,λ2,,λr such that λ1+λ2++λr=1 the vector


is in S.

Examples of convex sets in the plane are circles, triangles, and ellipses. The definition given above can be generalized to any real vector space:

Let V be a vector spaceMathworldPlanetmath (over or ). A subset S of V is convex if for all points x,y in S, the line segment {αx+(1-α)yα(0,1)} is also in S.

More generally, the same definition works for any vector space over an ordered field.

A polyconvex set is a finite union of compact, convex sets.

Remark. The notion of convexity can be generalized to an arbitrary partially ordered setMathworldPlanetmath: given a poset P (with partial ordering ), a subset C of P is said to be convex if for any a,bC, if cP is between a and b, that is, acb, then cC.

Title convex set
Canonical name ConvexSet
Date of creation 2013-03-22 11:46:35
Last modified on 2013-03-22 11:46:35
Owner drini (3)
Last modified by drini (3)
Numerical id 20
Author drini (3)
Entry type Definition
Classification msc 52A99
Classification msc 16G10
Classification msc 11F80
Classification msc 22E55
Classification msc 11A67
Classification msc 11F70
Classification msc 06A06
Synonym convex
Related topic ConvexCombination
Related topic CaratheodorysTheorem2
Related topic ExtremeSubsetOfConvexSet
Related topic PropertiesOfExtemeSubsetsOfAClosedConvexSet
Defines polyconvex set
Defines polyconvex