A Note on Koldobsky's Lattice Slicing Inequality

TL;DR

This paper proves a lower bound on the proportion of lattice points in an origin-symmetric convex body that lie in a hyperplane, partially addressing a question posed by Koldobsky.

Contribution

It establishes a new lattice slicing inequality for convex bodies, providing a quantitative bound involving volume and dimension.

Findings

Existence of a lattice hyperplane with a significant intersection with the convex body.

The bound depends inversely on the dimension and the volume of the body.

Partial resolution to Koldobsky's lattice slicing question.

Abstract

$ \newcommand{\R}{{\mathbb{R}}} \newcommand{\Z}{{\mathbb{Z}}} \renewcommand{\vec}[1]{{\mathbf{#1}}} $We show that if $K \subset \R^d$ is an origin-symmetric convex body, then there exists a vector $\vec{y} \in \Z^d$ 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.