| Popis: |
Let K be a centered convex body of volume 1 in R n . A direction θ ∈ S n − 1 is called sub-Gaussian for K with constant b > 0 if ‖ 〈 ⋅ , θ 〉 ‖ L ψ 2 ( 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 〉 ‖ L ψ 2 ( K ) ⩽ C ( log n ) 3 / 2 max { log n , a } L K for all θ in a subset Θ a of S n − 1 with σ ( Θ a ) ⩾ 1 − n − a , where C > 0 is an absolute constant. |
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