Extremal properties of central half-spaces for product measures

We deal with the isoperimetric and the shift problem for subsets of measure 1/2 in product probability spaces. We prove that the canonical central hall-spaces are extremal in particular cases: products of log-concave measures on the real line satisfying precise conditions and products: of uniform measures on spheres, or balls. As a corollary. we improve the known log-Sobolev constants for Euclidean balls. We also give some neu results about the related question of estimating the volume of sections of unit balls of f(p)-sums of Minkowski spaces. (C) 2001 Academic Press.