An example related to the slicing inequality for general measures

Abstract

For $n\in \mathbb{N}$ let $S_n$ be the smallest number $S>0$ satisfying the inequality $$ \int_K f \le S \cdot |K|^{\frac 1n} \cdot \max_{\xi\in S^{n-1}} \int_{K\cap \xi^\bot} f $$ for all centrally-symmetric convex bodies $K$ in $\mathbb{R}^n$ and all even, continuous probability densities $f$ on $K$. Here $|K|$ is the volume of $K$. It was proved by the second-named author that $S_n\le 2\sqrt{n}$, and in analogy with Bourgain's slicing problem, it was asked whether $S_n$ is bounded from above by a universal constant. In this note we construct an example showing that $S_n\ge c\sqrt{n}/\sqrt{\log \log n},$ where $c > 0$ is an absolute constant. Additionally, for any $0 < \alpha < 2$ we describe a related example that satisfies the so-called $\psi_{\alpha}$-condition.