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 half-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 new results about the related question of estimating the volume of sections of unit balls of ℓp-sums of Minkowski spaces.