2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161664The normalised volume measure on the $\ell_p^n$ unit ball ($1\leq p\leq 2$) satisfies the following isoperimetric inequality: the boundary measure of a set of measure $a$ is at least $cn^{1/p}\tilde{a}\log^{1-1/p}(1/\tilde{a})$, 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)ProbabilityMetric Geometrytext