Geodesic
Geometric stability via information theory
Discrete analysis (2016) Cet article a éte moissonné depuis la source Scholastica

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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 $\mathbb{Z}^d$ 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.
Publié le : 
@article{DAS_2016_a9,
     author = {David Ellis and Ehud Friedgut and Guy Kindler and Amir Yehudayoff},
     title = {Geometric stability via information theory},
     journal = {Discrete analysis},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2016_a9/}
}
TY  - JOUR
AU  - David Ellis
AU  - Ehud Friedgut
AU  - Guy Kindler
AU  - Amir Yehudayoff
TI  - Geometric stability via information theory
JO  - Discrete analysis
PY  - 2016
UR  - http://geodesic.mathdoc.fr/item/DAS_2016_a9/
LA  - en
ID  - DAS_2016_a9
ER  - 
%0 Journal Article
%A David Ellis
%A Ehud Friedgut
%A Guy Kindler
%A Amir Yehudayoff
%T Geometric stability via information theory
%J Discrete analysis
%D 2016
%U http://geodesic.mathdoc.fr/item/DAS_2016_a9/
%G en
%F DAS_2016_a9
David Ellis; Ehud Friedgut; Guy Kindler; Amir Yehudayoff. Geometric stability via information theory. Discrete analysis (2016). http://geodesic.mathdoc.fr/item/DAS_2016_a9/