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

Sodin, Sasha

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 362-373.

Nous prouvons une inégalité isopérimétrique pour la mesure uniforme $V_{p,n}$ sur la boule unité de ${\ell }_p^n(1\le p\le 2)$. Si $V_{p,n}\left(A\right)=a$, alors $V_{p,n}^{+}\left(A\right)\ge c{n}^{1/p}\tilde{a}{log}^{1-1/p}1/\tilde{a}$, où $V_{p,n}^{+}$ est la mesure de surface associée à $V_{p,n},\tilde{a}=min(a,1-a)$ et $c$ est une constante absolue. En particulier, les boules unités de ${\ell }_p^n$ vérifient la conjecture de Kannan-Lovász-Simonovits (Discrete Comput. Geom. 13 (1995)) sur la constante de Cheeger d'un corps convexe isotrope. La démonstration s'appuie sur les inégalités isopérimétriques de Bobkov (Ann. Probab. 27 (1999)) et de Barthe-Cattiaux-Roberto (Rev. Math. Iberoamericana 22 (2006)), et utilise la représentation de $V_{p,n}$ établie par Barthe-Guédon-Mendelson-Naor (Ann. Probab. 33 (2005)) ainsi qu'un argument de découpage.

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 $\tilde{a}=min(a,1-a)$.

Classification : 60E15, 28A75
Mots clés : isoperimetric inequalities, volume measure

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