Gruber P. Convex and Discrete Geometry

31 Covering with Convex Bodies 457
x ∈ C + ln. Hence (ln) is a bounded sequence in E d . By cancellation and renumbering,
if necessary, we may assume that l1, l2, ··· → l ∈ E d , say. Then l ∈ L by (6).
The inclusion x ∈ C + ln and the fact that C is closed yield x ∈ C + l. The proof of
(7) is complete.
Concluding, L is a covering lattice of C, by (7), and Propositions (5) and (1)
imply that
δ(C, L) =
V (C)
d(L)
V (C)
= lim
n→∞ d(Ln) = ϑL(C). ⊓⊔
Remark. A different proof shows that ϑT (C) is attained for suitable discrete sets T ,
see Hlawka [510] and Groemer [399].
The Covering Criterion of Wills
Given a convex body C in E d , when does a lattice provide a covering of C? There
is a small number of pertinent results. We cite the following interesting theorem of
Wills [1026].
Theorem 31.2. Let C be a proper convex body in Ed with V (C) > 1 2 S(C). Then the
integer lattice Zd is a covering lattice of C.
Remark. A generalization of this result which deals with multiple coverings is due
to Bokowski, Hadwiger and Wills [138]. For other covering criteria, see [447].
Considering the covering criterion, the following problem arises.
Problem 31.1. Extend the covering criterion to general lattices.
31.2 Star Numbers
Given a lattice covering of a convex body C in E d , its star number is the number of
the translates of C by lattice vectors, including C, which intersect the body C.
In the following we give a lower bound for star numbers due to Erdös and Rogers.
An upper bound in the spirit of the lower estimate of Swinnerton-Dyer for neighbours
in packings still seems to be missing. We formulate this as a problem:
Problem 31.2. Find a tight upper bound for the star number of lattice coverings of
(symmetric or general) proper convex bodies which (locally) have minimum density.
Lower Estimate for the Star Number
The precise lower bound for star numbers of lattice coverings of o-symmetric convex
bodies is due to Erdös and Rogers [308]:
Theorem 31.3. Let C be a proper, o-symmetric convex body and L a covering lattice
of C in E d . Then the star number of the covering {C + l : l ∈ L} is at least 2 d+1 − 1.