  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.
