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 \[ |K\cap \mathbb {Z}^n| \leq d_n\,|K\cap H\cap \mathbb {Z}^n|\,vol(K)^{\frac 1n}. \] 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.