Upper Bounds on Covering Minima of Convex Bodies

TL;DR

This paper introduces two new upper bounds on the covering minima of convex bodies, enhancing understanding of lattice coverings and improving bounds related to a conjecture on lattice polytopes.

Contribution

The paper presents novel upper bounds on covering minima that depend on projections and intersections, generalizing previous results and addressing open conjectures.

Findings

Bounds are sharp for direct sums of convex bodies.

Application to terminal simplices narrows bounds in a key conjecture.

Provides new insights into the maximal covering radius of 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 Gonzal\'ez 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.