A note on subgaussian estimates for linear functionals on convex bodies

Giannopoulos, A.; Pajor, A.; Paouris, G.

doi:10.1090/S0002-9939-07-08778-3

A note on subgaussian estimates for linear functionals on convex bodies
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by A. Giannopoulos, A. Pajor and G. Paouris PDF

Proc. Amer. Math. Soc. 135 (2007), 2599-2606 Request permission

Abstract:

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If $K$ is a convex body in ${\mathbb R}^n$ with volume one and center of mass at the origin, there exists $x\neq 0$ such that \[ |\{ y\in K: |\langle y,x\rangle |\geq t\|\langle \cdot ,x\rangle \|_1\}|\leq \exp (-ct^2/\log ^2(t+1))\] for all $t\geq 1$, where $c>0$ is an absolute constant. The proof is based on the study of the $L_q$–centroid bodies of $K$. Analogous results hold true for general log-concave measures.

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