Majorization of Gaussian processes and geometric applications

SummaryLet (xi*)i=1n denote the decreasing rearrangement of a sequence of real numbers (xi)i=1n. Then for everyi≠j, and every 1≦k≦n, the 2-nd order partial distributional derivatives satisfy the inequality, $$\frac{{\partial ^2 }}{{\partial x_i \partial x_j }}\left( {\sum\limits_{l = 1}^k {x_l^* } } \right) \leqq 0$$ . As a consequence we generalize the theorem due to Fernique and Sudakov. A generalization of Slepian's lemma is also a consequence of another differential inequality. We also derive a new proof and generalizations to volume estimates of intersecting spheres and balls in ℝn proved by Gromov.