Sub-Gaussian directions of isotropic convex bodies

Abstract

Let K be a centered convex body of volume 1 in Rn. A direction θ∈Sn−1 is called sub-Gaussian for K with constant b>0 if ‖〈⋅,θ〉‖Lψ2(K)⩽b‖〈⋅,θ〉‖2. We show that if K is isotropic then most directions are sub-Gaussian with a constant which is logarithmic in the dimension. More precisely, for any a>1 we have‖〈⋅,θ〉‖Lψ2(K)⩽C(log⁡n)3/2max⁡{log⁡n,a}LK for all θ in a subset Θa of Sn−1 with σ(Θa)⩾1−n−a, where C>0 is an absolute constant.