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