Reducing the Large Set Threshold for Oertel's Conjecture on the Mixed-Integer Volume

TL;DR

This paper improves the understanding of Oertel's conjecture for mixed-integer convex sets by reducing the size threshold needed for the conjecture to hold, from exponential to polynomial in the dimension.

Contribution

It introduces a geometric approach that significantly lowers the size threshold for the conjecture, expanding the class of sets where it applies.

Findings

Threshold reduced from exponential to polynomial in dimension

Broader class of mixed-integer convex sets satisfy the conjecture

Enhanced geometric methods for analyzing convex sets

Abstract

In 1960, Gr\"{u}nbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently, in 2014, Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, he proposed that for any convex body $C\subset \mathbb{R}^{n+d}$, there should exist a point $\mathbf{x} \in S=C\cap(\mathbb{Z}^{n}\times\mathbb{R}^d)$ such that for every halfspace $H$ containing $\mathbf{x}$, one has that \[ \mathcal{H}_d(H\cap S) \geq \frac{1}{2^n}\frac{1}{e}\mathcal{H}_d(S), \] where $\mathcal{H}_d$ denotes the $d$-dimensional Hausdorff measure. While the conjecture remains open, Basu and Oertel proved in 2017 that the above inequality holds true for sufficiently large sets, in terms of a measure known as the \emph{lattice width} of a set. In this work, by following a geometric approach, we improve this result by substantially reducing the threshold at which a set can be considered large. We reduce this threshold from an exponential to a polynomial dependency on the dimension, therefore significantly enlarging the family of mixed-integer convex sets over which Oertel's conjecture holds true.