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

Andrés Cristi, David Salas

TL;DR

This paper tackles Oertel's conjecture for the mixed-integer volume by establishing polynomial-threshold conditions under which the conjectured bound holds. The authors combine a slice-based volume analysis, centroid shifting to integrate over the mixed-integer lattice, and a Grünbaum-type bound to propagate continuous-volume estimates to the mixed-integer setting. The main results show that a ball of radius $k$ in the projection, with $k\ge\alpha d^2 n^{3/2}$, guarantees the bound $\mathcal{F}(S)\ge\frac{1}{2^n}\left(\frac{d}{d+1}\right)^d$, and a unimodular-transform argument yields a lattice-width threshold $\omega(\mathrm{proj}_{\mathbb{R}^n}(C))\ge\alpha' d^2 n^6$ with the same conclusion; a sharper $n=1$ bound improves the threshold to $k\ge\alpha'' d$. These polynomial thresholds substantially broaden the class of mixed-integer convex sets for which Oertel's conjecture holds, offering strong evidence toward its general validity.

Abstract

In 1960, Grünbaum 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.

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

TL;DR

This paper tackles Oertel's conjecture for the mixed-integer volume by establishing polynomial-threshold conditions under which the conjectured bound holds. The authors combine a slice-based volume analysis, centroid shifting to integrate over the mixed-integer lattice, and a Grünbaum-type bound to propagate continuous-volume estimates to the mixed-integer setting. The main results show that a ball of radius in the projection, with , guarantees the bound , and a unimodular-transform argument yields a lattice-width threshold with the same conclusion; a sharper bound improves the threshold to . These polynomial thresholds substantially broaden the class of mixed-integer convex sets for which Oertel's conjecture holds, offering strong evidence toward its general validity.

Abstract

In 1960, Grünbaum proved that for any convex body and every halfspace containing the centroid of , one has that the volume of is at least a -fraction of the volume of . Recently, in 2014, Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, he proposed that for any convex body , there should exist a point such that for every halfspace containing , one has that where denotes the -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.

Paper Structure

This paper contains 17 sections, 12 theorems, 60 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A worst-case-like example and Oertel's conjecture on the mixed-integer volume:
  3. Partial results on Oertel's conjecture:
  4. Our contributions:
  5. Preliminaries
  6. Some facts about cones
  7. Oertel radius
  8. The case $n=1$
  9. Technical Lemmas for the case $n=1$
  10. Main result for the case $n=1$
  11. The general case
  12. Technical Lemmas for the general case
  13. Main result for the general case
  14. A new Threshold for the Lattice Width
  15. Final comments and perspectives
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a universal constant $\alpha>0$ such that for every $n,d\in\mathbb{N}$ and every convex body $C\subset\mathbb{R}^{n+d}$ with $\omega(\mathop{\rm proj}\nolimits_{\mathbb{R}^n}(C)) > 2cn(n+d)^{5/2}\alpha n^{n+1}$ for some $c\in\mathbb{R}_+$, then In particular, if $c\in\mathbb{R}_+$ is such that $e^{-\frac{1}{c}-1} +e^{-\frac{2}{c}} - 1\geq 2^{-(n+1)}$, then Conjecture conj:Oertel hold

Figures (8)

  • Figure 1: Estimation of $\bar{c}$
  • Figure 2: Subcone $K' = \mathop{\rm cone}\nolimits(x,B')$ of $K' = \mathop{\rm cone}\nolimits(x,B')$ with hight $h'<h$.
  • Figure 3: Illustration that $L\subset C$ and $C_i \subset ((i+1)/i)\cdot L$.
  • Figure 4: Illustrations of inner and outer approximations from the proof of \ref{['lem:upper_and_lower_bound_n_1']}. The section $C_{\lfloor k/2\rfloor}$ is approximated twice in the inner approximation or is missing in the outer approximation.
  • Figure 5: Illustration of the modification of the centroid after cutting a part of $C$. The centroid of $C_R^w$ is to the left of the centroid of $C_R$. Accordingly, the new centroid of $C^w$ is also moved to the left.
  • ...and 3 more figures

Theorems & Definitions (24)

  • Conjecture 1: Oertel's conjecture oertel2014integer
  • Theorem 1.1: BasuOertel2017Centerpoints
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 14 more