Uniform almost sub-Gaussian estimates for linear functionals on convex sets

Abstract

A well-known consequence of the Brunn-Minkowski inequality, is that the distribution of a linear functional on a convex set has a sub-exponential tail. That is, for any dimension n, a convex set K ⊂ Rn of volume one, and a linear functional φ : Rn → R, we have V oln ({ x ∈ K; |φ(x)| > t‖φ‖L1(K) }) ≤ e−ct for all t > 1, where ‖φ‖L1(K) = ∫ K |φ(x)|dx and c > 0 is a universal constant. In this note we prove that for any dimension n and a convex set K ⊂ Rn of volume one, there exists a non-zero linear functional φ : Rn → R such that V oln ({ x ∈ K; |φ(x)| > t‖φ‖L1(K) }) ≤ e t2 log5(t+1) for all t > 1, where c > 0 is a universal constant.