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

Theorems & Definitions (10)

• Theorem 1.1
• Corollary 1.2
• Theorem 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm:affine']}
• Proposition 4.1
• proof : Proof of Proposition \ref{['prop:deviation']}
• proof : Proof of Theorem \ref{['thm:centered']}
• proof : Proof of Corollary \ref{['cor:central']}