We show that given a symmetric convex set KCR d ; the function t-gðe t KÞ is log-concave on R; where g denotes the standard d-dimensional Gaussian measure. We also comment on the extension of this property to unconditional log-concave measures and sets, and on the complex case. r

We establish new functional versions of the Blaschke-Santaló inequality on the volume product of a convex body which generalize to the nonsymmetric setting an inequality of K. Ball [2] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for logconcave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman [1], with its equality case.

No abstract