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

## TL;DR

This paper investigates upper bounds on the covering radius of non-hollow lattice polytopes, proposing a conjecture, proving it in low dimensions, and connecting it to a discrete surface area concept inspired by Hadwiger's formula.

## Contribution

It introduces a new discrete surface area for simplices, proves an upper bound conjecture in dimensions up to three, and links the problem to a discrete analog of Hadwiger's formula.

## Key findings

- Conjecture that the covering radius is at most d/2 for certain simplices
- Proved the conjecture for dimensions up to three
- Introduced and validated a discrete surface area concept

## 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 Gonz\'alez-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.