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Upper Bounds on Covering Minima of Convex Bodies

Katarina Krivokuća

TL;DR

The paper advances the study of covering minima by introducing two upper-bounds based on projections and intersections, with a sharp direct-sum equality that unifies bounds for the covering radius and lattice width. It further shows that the intersection-bound is tight for locally anti-blocking bodies and derives corollaries that connect to terminal polytopes. By applying these results to standard terminal simplices and their weighted variants, the authors tighten upper bounds on $\mu_i(T_d)$ and illuminate the relationship between terminal simplices and terminal polytopes, ultimately linking key conjectures in the area. The work also strengthens the bridge between projections, intersections, and computable covering radii, contributing to a deeper understanding of maximal covering radii for non-hollow lattice polytopes.

Abstract

We give two new upper bounds on the covering minima of convex bodies, depending on covering minima of certain projections and intersections with linear subspaces. We show one bound to be sharp for direct sums of two convex bodies, generalizing previous results on the covering radius and lattice width of direct sums. We apply our results to standard terminal simplices, reducing the gap between the upper and lower bounds in a conjecture of Gonzaléz Merino and Schymura (2017), which gives insight on a conjecture of Codenotti, Santos and Schymura (2021) on the maximal covering radius of a non-hollow lattice polytope.

Upper Bounds on Covering Minima of Convex Bodies

TL;DR

The paper advances the study of covering minima by introducing two upper-bounds based on projections and intersections, with a sharp direct-sum equality that unifies bounds for the covering radius and lattice width. It further shows that the intersection-bound is tight for locally anti-blocking bodies and derives corollaries that connect to terminal polytopes. By applying these results to standard terminal simplices and their weighted variants, the authors tighten upper bounds on and illuminate the relationship between terminal simplices and terminal polytopes, ultimately linking key conjectures in the area. The work also strengthens the bridge between projections, intersections, and computable covering radii, contributing to a deeper understanding of maximal covering radii for non-hollow lattice polytopes.

Abstract

We give two new upper bounds on the covering minima of convex bodies, depending on covering minima of certain projections and intersections with linear subspaces. We show one bound to be sharp for direct sums of two convex bodies, generalizing previous results on the covering radius and lattice width of direct sums. We apply our results to standard terminal simplices, reducing the gap between the upper and lower bounds in a conjecture of Gonzaléz Merino and Schymura (2017), which gives insight on a conjecture of Codenotti, Santos and Schymura (2021) on the maximal covering radius of a non-hollow lattice polytope.
Paper Structure (8 sections, 14 theorems, 51 equations, 1 figure)

This paper contains 8 sections, 14 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Our Contributions
  4. Organization of the paper
  5. Projection Bound on Covering Minima
  6. Intersection Bound on Covering Minima
  7. Covering radius of non-hollow lattice polytopes
  8. Upper Bounds on Covering Minima of (Weighted) Terminal Simplices

Key Result

Theorem 1.1

Let $K\subseteq\mathbb{R}^d$ be a convex body containing the origin, $\Lambda\subseteq \mathbb{R}^d$ a lattice and $V\subseteq \mathbb{R}^d$ a rational linear subspace, with $\dim V=\ell$ and $i\in[d]$. If $\pi_V$ denotes the orthogonal projection of $\mathbb{R}^d$ to $V$, the following inequality h

Figures (1)

  • Figure 1: Dependencies of our bounds

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof : Proof of Theorem \ref{['thm: ub proj']}
  • proof : Proof of Theorem \ref{['thm: direct sum']}
  • Example 2.2
  • proof : Proof of Theorem \ref{['thm: ub intersec']}
  • Definition 3.2
  • Corollary 3.3
  • ...and 19 more