Uniform almost sub-Gaussian estimates for linear functionals on convex sets

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

$$\operatorname{Vol}_n (\lbrace x \in K; \vert\varphi(x)\vert > t \Vert \varphi \Vert _{L_1(K)}\rbrace) \le e^{- c t} \enskip$$ for all $$\enskip t > 1,$$

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

$$\operatorname{Vol}_n(\lbrace x \in K; \vert\varphi(x)\vert > t \Vert \varphi \Vert _{L_1(K)}\rbrace) \le e^{- c \frac{t^2}{\log^5 (t+1)} } \enskip$$ for all $$\enskip t > 1,$$

where $c > 0$ is a universal constant.