Generalized dual Sudakov minoration via dimension-reduction - A program

We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures μ, of the classical dual Sudakov minoration on the expectation of the supremum of a Gaussian process: (Equation Presented) Here K is an origin-symmetric convex body, Z_p(μ) is the L_p-centroid body associated to μ, M(A,B) is the packing number of B in A, and C > 0 is a universal constant. The program is to first establish a weak generalized dual Sudakov minoration, involving the dimension n of the ambient space, which is then self-improved to a dimension-free estimate after applying a dimension-reduction step. The latter step may be thought of as a conjectural "small-ball one-sided" variant of the Johnson{Lindenstrauss dimension-reduction lemma. We establish the weak generalized dual Sudakov minoration for a variety of log-concave probability measures and convex bodies (for instance, this step is fully resolved assuming a positive answer to the slicing problem). The separation dimension-reduction step is fully established for ellipsoids and, up to logarithmic factors in the dimension, for cubes, resulting in a corresponding generalized (regular) dual Sudakov minoration estimate for these bodies and arbitrary log-concave measures, which are shown to be (essentially) best possible. Along the way, we establish a regular version of (0.1) for all p ≥ n and provide a new direct proof of Sudakov minoration via the program.