Polynomial Bounds in Koldobsky's Discrete Slicing Problem

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 $c\,n^2\,\omega(n)$, where $c$ is an absolute constant and $\omega(n)$ is the flatness constant. Due to the best known upper bound on $\omega(n)$ this gives a ${c\,n^{10/3}\log(n)^a}$ bound on $d_n$ where $a$ is another absolute constant. This bound improves on former bounds which were exponential in the dimension