Geometric stability via information theory

David Ellis, Ehud Friedgut, Guy Kindler, Amir Yehudayoff

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. In this paper, we prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close in symmetric difference to a box. Our results are best possible up to a constant factor depending upon the dimension alone. Our approach is information theoretic. We use our stability result for the Loomis-Whitney inequality to obtain a stability result for the edge-isoperimetric inequality in the infinite d-dimensional lattice. Namely, we prove that a subset of ℤd with small edge-boundary must be close in symmetric difference to a d-dimensional cube. Our bound is, again, best possible up to a constant factor depending upon d alone.

Original languageAmerican English
Pages (from-to)1-28
Number of pages28
JournalDiscrete Analysis
Volume10
Issue number2016
DOIs
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© 2016 David Ellis, Ehud Friedgut, Guy Kindler and Amir Yehudayoff.

Keywords

  • Entropy
  • Information theory
  • Loomis-Whitney inequality
  • Projections
  • Stability

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