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 nge 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.
Highlights
The covering radius of a convex body K in Rd with respect to a lattice is defined as μ(K, ) = min {μ ≥ 0 : μK + = Rd }
The covering radius is a classical parameter in the Geometry of Numbers, in particular in the realm of transference results, the reduction of quadratic forms, and Diophantine Approximations
Translations and/or direct sums of the S(1l )s produce nine pairwise non-equivalent non-hollow lattice 3-polytopes of covering radius 3/2, that we describe in Lemma 3.8
Summary
Translations and/or direct sums of the S(1l )s produce nine pairwise non-equivalent non-hollow lattice 3-polytopes of covering radius 3/2, that we describe in Lemma 3.8. The sum of these numbers is a version of the “surface area” of S, except that the volume of each facet is computed with respect to the lattice projected from the opposite vertex Motivated by this definition and Theorem 1.4 we propose the following conjecture, which is the main object of study in this paper: Conjecture C Let S be a d-simplex with the origin in its interior and with rational vertex directions.
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