A Quantitative Version of Carathéodory’s Theorem for Convex Sets

  • Reinhold Meise
  • Alan Taylor
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 
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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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