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 and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every $$n\ge d$$ 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 give strong evidence for its validity in arbitrary dimension.

Computing the Covering Radius of a Polytope with an Application to Lonely Runners

“…The intuition behind this notion is that if we consider the lattice arrangements µK + Z n , for increasing values of µ ≤ μ, then p does not belong to any lattice translate of µK unless µ = μ. This concept is very natural for the investigation of the covering radius and already appeared before, for instance in [8]. Figure 1 illustrates the concept of last-covered points on a covering by triangles and another one by squares.…”

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

“…The intuition behind this notion is that if we consider the lattice arrangements µK + Z n , for increasing values of µ ≤ μ, then p does not belong to any lattice translate of µK unless µ = μ. This concept is very natural for the investigation of the covering radius and already appeared before, for instance in [8].…”