Sudakov Minoration for Unconditional Log-Concave Vectors

Abstract

Sudakov minoration asks whether a large family of separated random linear forms must have a large expected maximum. We present a proof candidate for this principle for all unconditional log-concave vectors. The argument starts with a classical reduction to sparse coordinate supports. It then selects one feasible threshold witness for each label, removes a common coordinate core, and controls the remaining overlaps. The main analytic step is an exponential-moment estimate for a softened witness payoff. We obtain it by combining a one-dimensional clipping inequality with triangular transport, using log-concavity of the source and sign symmetry of the target. A Bernoulli comparison restores the signs at the end. The exposition includes historical context, a guide to the proof, all intermediate arguments, and explicit choices of constants. Two consequences concern covering numbers in the moment metric and concentration of bounded functions of the magnitudes.