Geodesic
Uniform almost sub-Gaussian estimates for linear functionals on convex sets
Algebra i analiz, Tome 19 (2007) no. 1, pp. 93-108.

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A well-known consequence of the Brunn–Minkowski inequality says that the distribution of a linear functional on a convex set has a uniformly subexponential tail. That is, for any dimension $n$, any convex set $K\subset \mathbb{R}^n$ of volume one, and any linear functional $\varphi:\mathbb{R}^n\rightarrow \mathbb{R}$, we have $$ \operatorname{Vol}_n(\lbrace x\in K;\vert\varphi(x)\vert>t\Vert\varphi\Vert _{L_1(K)}\rbrace) \le e^{-ct}\enskip \text{for all }t>1, $$ where $\Vert \varphi\Vert _{L_1(K)}=\int_K\vert\varphi(x)\vert d x$ and $c>0$ is a universal constant. In this paper, it is proved that for any dimension $n$ and a convex set $K\subset\mathbb{R}^n$ of volume one, there exists a nonzero linear functional $\varphi:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $\displaystyle\operatorname{Vol}_n(\lbrace x\in K;\vert\varphi(x)\vert>t\Vert\varphi\Vert _{L_1(K)}\rbrace) \le e^{-c\frac{t^2}{\log^5 (t+1)}}\enskip$ for all $\displaystyle\enskip t>1,$ where $c>0$ is a universal constant.
Keywords: Hyperbolic dimension, Gromov's asymptotic dimension. 
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S.. Buyalo; V. Shroeder. Uniform almost sub-Gaussian estimates for linear functionals on convex sets. Algebra i analiz, Tome 19 (2007) no. 1, pp. 93-108. http://geodesic.mathdoc.fr/item/AA_2007_19_1_a4/

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