Polynomial Bounds in Koldobsky's Discrete Slicing Problem

TL;DR

This paper establishes polynomial upper bounds on a constant related to Koldobsky's discrete slicing problem, improving previous exponential bounds and connecting geometric and lattice point properties of convex bodies.

Contribution

The authors provide the first polynomial bounds on the constant in Koldobsky's problem, advancing understanding of lattice points in convex bodies and their sections.

Findings

Bound d_n by c n^2 ω(n)/log(n)

Derived a c n^3 log(n)^2 bound using recent flatness constant bounds

Improved previous exponential bounds to polynomial in dimension

Abstract

In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace $H\subset\mathbb{R}^n$ with \[ |K\cap\mathbb Z^n| \leq d_n\,|K\cap H\cap \mathbb Z^n|\,\mathrm{vol}(K)^{\frac 1n}. \] In this article we show that $d_n$ is bounded from above by $c\,n^2\,\omega(n)/\log(n)$, where $c$ is an absolute constant and $\omega(n)$ is the flatness constant. Due to the recent best known upper bound on $\omega(n)$ we get a ${c\,n^3\log(n)^2}$ bound on $d_n$. This improves on former bounds which were exponential in the dimension.