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 $c{n}^2\omega (n)/\log (n)$, 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.