## The Annals of Probability

### Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures

S. G. Bobkov

#### Abstract

We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincaré-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.

#### Article information

Source
Ann. Probab. Volume 27, Number 4 (1999), 1903-1921.

Dates
First available in Project Euclid: 31 May 2002

http://projecteuclid.org/euclid.aop/1022874820

Digital Object Identifier
doi:10.1214/aop/1022874820

Mathematical Reviews number (MathSciNet)
MR1742893

Zentralblatt MATH identifier
0964.60013

Subjects
Primary: 60J75
Secondary: 60G09: Exchangeability 60G57: Random measures 05A18: Partitions of sets

#### Citation

Bobkov, S. G. Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures. Ann. Probab. 27 (1999), no. 4, 1903--1921. doi:10.1214/aop/1022874820. http://projecteuclid.org/euclid.aop/1022874820.

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