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Isoperimetric problems for convex bodies and a localization lemma

Authors:

Affiliation:

1. Department of Computer Science, Carnegie-Mellon University, 15213, Pittsburgh, PA, USA
2. Department of Computer Science, Yale University, 06520, New Haven, CT, USA
3. Mathematical Institute, Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053, Budapest, Hungary

Abstract:

We study the smallest number ψ(K) such that a given convex bodyK in ℝⁿ can be cut into two partsK ₁ andK ₂ by a surface with an (n−1)-dimensional measure ψ(K) vol(K ₁)·vol(K ₂)/vol(K). LetM ₁(K) be the average distance of a point ofK from its center of gravity. We prove for the “isoperimetric coefficient” that

$psi (K) geqslant frac{{ln 2}}{{M_1 (K)}}$

Keywords:
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