The Mathematical Legacy of Leon Ehrenpreis pp 233-245 | Cite as

# A Quantitative Version of Carathéodory’s Theorem for Convex Sets

## Abstract

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.

## Keywords

Induction Hypothesis Extreme Point Interior Point Convex Combination Compact Convex## Notes

### Acknowledgement

We wish to thank Alexander Barvinok for several helpful conversations about this work.

## References

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