A discrete version of Koldobsky's slicing inequality

TL;DR

This paper establishes a discrete analogue of Koldobsky's slicing inequality for lattice points in convex bodies, providing bounds involving volume and lattice intersections, with constants depending on dimension and body symmetry.

Contribution

It introduces a discrete version of Koldobsky's slicing inequality with explicit bounds on lattice points, depending on dimension and symmetry properties.

Findings

Bound on lattice points in convex bodies involving volume and subspace intersections

Constant C(d) can be chosen with specific asymptotic growth rates

Unconditional bodies allow for improved bounds on C(d)

Abstract

Let $\# K$ be a number of integer lattice points contained in a set $K$. In this paper we prove that for each $d\in {\mathbb N}$ there exists a constant $C(d)$ depending on $d$ only, such that for any origin-symmetric convex body $K \subset {\mathbb R}^d$ containing $d$ linearly independent lattice points $$ \# K \leq C(d)\max(\# (K\cap H))\, {\rm vol}_d(K)^{\frac{d-m}{d}}, $$ where the maximum is taken over all $m$-dimensional subspaces of ${\mathbb R}^d$. We also prove that $C(d)$ can be chosen asymptotically of order $O(1)^{d}d^{d-m}$. In addition, we show that if $K$ is an unconditional convex body then $C(d)$ can be chosen asymptotically of order $O(d)^{d-m}$.