Helly’s theorem

Suppose A1,,Amd is a family of convex sets, and every d+1 of them have a non-empty intersectionMathworldPlanetmathPlanetmath. Then i=1mAi is non-empty.


The proof is by inductionMathworldPlanetmath on m. If m=d+1, then the statement is vacuousPlanetmathPlanetmath. Suppose the statement is true if m is replaced by m-1. The sets Bj=ijAi are non-empty by inductive hypothesis. Pick a point pj from each of Bj. By Radon’s lemma, there is a partition of p’s into two sets P1 and P2 such that I=(convP1)(convP2). For every Aj either every point in P1 belongs to Aj or every point in P2 belongs to Aj. Hence IAj for every j. ∎

Title Helly’s theoremMathworldPlanetmath
Canonical name HellysTheorem
Date of creation 2013-03-22 13:57:38
Last modified on 2013-03-22 13:57:38
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 6
Author bbukh (348)
Entry type Theorem
Classification msc 52A35