A Note on Koldobsky's Lattice Slicing Inequality

Abstract

$ \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \renewcommand{\vec}[1]{\mathbf{#1}} $We show that if $K\subset {\mathbb{R}}^d$ is an origin-symmetric convex body, then there exists a vector $\vec{y}\in {\mathbb{Z}}^d$ such that \begin{align*} |K \cap \Z^d \cap \vec{y}^\perp| / |K \cap \Z^d| \ge \min(1,c \cdot d^{-1} \cdot \mathrm{vol}(K)^{-1/(d-1)}) \; , \end{align*} for some absolute constant $c>0$, where ${\vec{y}}^{\perp }$ denotes the subspace orthogonal to $\vec{y}$. This gives a partial answer to a question by Koldobsky.