The covering radius and a discrete surface area for non-hollow simplices

Abstract

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of $d/2$ in dimension $d$, achieved by the "standard terminal simplices" and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino \& Schymura (2017) that the $d$-th covering minimum of the standard terminal $n$-simplex equals $d/2$, for every $n>d$. We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger's formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and we give strong evidence for its validity in arbitrary dimension.

arXiv comment44 pages, 7 figures

References in corpus (5)

• Canonical toric Fano threefolds
• Classification of empty lattice $4$-simplices of width larger than two
• On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture
• Three-dimensional lattice polytopes with two interior lattice points
• Non-spanning lattice 3-polytopes

Cited by in corpus (1)

• Computing the covering radius of a polytope with an application to lonely runners