convex set
Let S a subset of ℝn. 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 equivalent 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
r∑k=1λkuk |
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 space (over ℝ or ℂ). A subset S of V
is convex if for all points x,y in S, the line segment
{αx+(1-α)y∣α∈(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 set: given a poset P (with partial ordering ≤), a subset C of P is said to be convex if for any a,b∈C, if c∈P is between a and b, that is, a≤c≤b, then c∈C.
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 |