An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies

Emanuel Milman; Sasha Sodin

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An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies

Emanuel Milman·Sasha Sodin

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Abstract

We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.

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Emanuel Milman, Sasha Sodin. 2007-11-21. An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies. https://arxiv.org/abs/math/0703857

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