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 \begin{equation*} \operatorname {Vol}_n (\lbrace x \in K; |\varphi (x)| > t \| \varphi \|_{L_1(K)}\rbrace ) \le e^{- c t} \quad \text {for all}\quad t > 1, \end{equation*} where $\| \varphi \|_{L_1(K)} = \int _K |\varphi (x)| d x$ 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 \begin{equation*} \operatorname {Vol}_n(\lbrace x \in K; |\varphi (x)| > t \| \varphi \|_{L_1(K)}\rbrace ) \le e^{- c \frac {t^2}{\log ^5 (t+1)} } \quad \text {for all}\quad t > 1, \end{equation*} where $c > 0$ is a universal constant.