A Quantitative Version of Carathéodory’s Theorem for Convex Sets

Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 16)


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 xK 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 from its boundary.


Induction 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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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  1. 1.Mathematisches InstitutHeinrich-Heine-UniversitätDüsseldorfGermany
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

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