The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices.

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 ≥ 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.