Sub-Gaussian directions of isotropic convex bodies

Let K   be a centered convex body of volume 1 in RnRn. A direction θ∈Sn−1θ∈Sn−1 is called sub-Gaussian for K   with constant b>0b>0 if ‖〈⋅,θ〉‖Lψ2(K)⩽b‖〈⋅,θ〉‖2‖〈⋅,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>1a>1 we have‖〈⋅,θ〉‖Lψ2(K)⩽C(log⁡n)3/2max⁡{log⁡n,a}LK for all θ   in a subset ΘaΘa of Sn−1Sn−1 with σ(Θa)⩾1−n−aσ(Θa)⩾1−n−a, where C>0C>0 is an absolute constant.