A Quantitative Version of Carathéodory’s Theorem for Convex Sets
Carathéodory’s theorem for compact convex sets K⊂ℝ m shows that every point x of K lies in the convex hull of m+1 extreme points of K; that is, in the m-simplex with vertices at m+1 extreme points. However, it need not be the case that if x is a positive distance away from the boundary of K, then x is a positive distance away from the boundary of one of these simplices. Here, we show that if K has only finitely many extreme points, then there are a finite set F⊂∂K and a constant c>0 such that if x∈K is of distance δ>0 from the boundary of K, then x belongs to one of the m-simplices with vertices from F and is of distance at least cδ from its boundary.
KeywordsInduction Hypothesis Extreme Point Interior Point Convex Combination Compact Convex
We wish to thank Alexander Barvinok for several helpful conversations about this work.
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