An isoperimetric inequality on the ${\ell }_p$ balls

Abstract

The normalised volume measure on the ${\ell }_p^n$ unit ball ($1\le p\le 2$) satisfies the following isoperimetric inequality: the boundary measure of a set of measure $a$ is at least $c{n}^{1/p}\tilde{a}{\log }^{1-1/p}(1/\tilde{a})$, where $\tilde{a}=\min (a,1-a)$.

Published in at http://dx.doi.org/10.1214/07-AIHP121 the Annales de l'Institut Henri Poincaré - Probabilités et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• A probabilistic approach to the geometry of the \ell_p^n-ball
• Between Sobolev and Poincaré

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