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 $\varphi\colon\mathbb{R}^n\to\mathbb{R}$, we have
$$\operatorname{Vol}_n(\{x\in K;|\varphi(x)|>t\|\varphi\|_{L_1(K)}\})\le e^{-ct}\quad \text{for all}\quad t>1,$$
where $\|\varphi\|_{L_1(K)}=\int_K|\varphi(x)|\,dx$ 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 $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ such that
$$\operatorname{Vol}_n(\{x\in K;|\varphi(x)|>t\|\varphi\|_{L_1(K)}\})\le e^{-c\frac{t^2}{\log^5 (t+1)}} \quad \text{for all}\quad t>1,$$
where $c>0$ is a universal constant.