On isoperimetric constants for log-concave probability distributions

Lower bounds on the isoperimetric constant for logarithmically concave probability measures are considered in terms of the distribution of the Euclidean norm. A refined form of Kannan-Lovasz-Simonovits' inequality is obtained. Given a Borel probability measure p on R-n, its isoperimetric constant or, isoperimetric coefficient, is defined as the optimal value h = h(mu) satisfying an isoperimetric-type inequality mu(+)(A) >= h min {mu(A), 1 - mu(A)} Here, A is an arbitrary Borel subset of R-n of measure mu(A) with p-perimeter mu(+) (A) = lim(epsilon down arrow 0) mu(A(epsilon))-mu(A)/epsilon, where A(epsilon) = {x epsilon R-n : vertical bar x - a vertical bar < epsilon, for some a epsilon A} denotes an open E-neighbourhood of A with respect to the Euclidean distance. The quantity h(mu) represents an important geometric characteristic of the measure and is deeply related to a number of interesting analytic inequalities. As an example, one may consider a Poincare-type inequality integral vertical bar del f vertical bar(2) d mu >= lambda(1)integral vertical bar f vertical bar(2)d mu in the class of all smooth functions f on R-n such that integral f d mu = 0. The optimal value lambda(1), the so called spectral gap, satisfies lambda(1) >= h(2)/4. This relation goes back to the work by J. Cheeger in the framework of Riemannian manifolds [C] and - in a more general form - to earlier works by V.G. Maz'ya (cf. [M1-2], [G]). The problem on bounding these two quantities from below has a long story. In this note we specialize to the class of log-concave probability measures, in which case, as was recently shown by A Ledoux [L], lambda(1) and h are equivalent (lambda(1) <= 36 h(2)).