Asymmetry of convex sets with isolated extreme points

Toth, Gabor

doi:10.1090/S0002-9939-08-09499-9

Asymmetry of convex sets with isolated extreme points
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by Gabor Toth

Proc. Amer. Math. Soc. 137 (2009), 287-295

DOI: https://doi.org/10.1090/S0002-9939-08-09499-9

Published electronically: June 30, 2008

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Abstract:

When measuring asymmetry of convex sets $\mathcal {L}\subset \mathbf {R} ^n$ in terms of inscribed simplices, the interior of $\mathcal {L}$ naturally splits into regular and singular sets. Based on examples, it may be conjectured that the singular set is empty iff $\mathcal {L}$ is a simplex. In this paper we prove this conjecture with the additional assumption that $\mathcal {L}$ has at least $n$ isolated extreme points on its boundary.

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