Extremal slabs in the cube and the Laplace transform

We study the volume of symmetric slabs in the unit cube. We show that, for t<3/4, the slab parallel to a face has the minimal volume among all symmetric slabs with width t. For large width, we prove the asymptotic extremality of the slab orthogonal to the main diagonal. The proof is based on certain concavity properties of the Laplace transform and on several limit theorems from probability: the central limit theorem and classical principles of moderate and large deviations. Finally, we extend some of the results to more general classes of bodies. (C) 2003 Elsevier Science (USA). All rights reserved.