Gruber P. Convex and Discrete Geometry

454 Geometry of Numbers
The fourth step is to show the following:
(11) Let E be a strictly convex disc. Then the upper density of a packing of
translates of E never exceeds the density of the densest lattice packing of E.
Let D = 1 2 (E − E). Similar arguments as in the proof of (10) but with E, D instead
of D, C yield (11), concluding the proof of the theorem for strictly convex discs.
For convex discs which are not strictly convex consider approximation with
strictly convex discs and use (11). The details are tedious. ⊓⊔
Remark. This result does not extend to packing by congruent copies, as the example
of a triangle shows. More precisely, for a typical convex disc C (in the sense of Baire
categories), Fejes Tóth [320] showed that there are packings of congruent copies of
C which have density larger than δL(C). It has been conjectured that this holds in all
dimensions d ≥ 2. For certain ellipsoids this was shown by Bezdek and Kuperberg
[110] and Rogers [851] conjectured it for Euclidean balls for all sufficiently large d.
In the special case of packings by congruent copies of centrally symmetric convex
discs, the theorem continues to hold according to a result of Fejes Tóth [327],
[329], p. 86, which is a consequence of one of his more general results.
A Conjecture of Zassenhaus for Densest Packings in E d
A periodic packing of a proper convex body C is a packing by translates, where the
set T of translation vectors is of the form
T = L ∪ (L + t1) ∪···∪(L + tm)
with a lattice L and vectors t1,...,tm ∈ E d . It is easy to see that, for every proper
convex body C, there are periodic packings with density arbitrarily close to δT (C).
An interesting open conjecture of Zassenhaus [1042] asserts even more:
Conjecture 30.2. Let C be a proper convex body in E d . Then δT (C) is attained by a
suitable periodic packing of C.
31 Covering with Convex Bodies
The theory of covering is less rich than the theory of packing with convex bodies
and appeared much later in the literature, the first landmark being a result of
Kershner [578]. It shows that the minimum density of coverings of E 2 by congruent
circular discs is attained by lattice coverings. One reason for the fact that covering
results have attracted less interest is that the arithmetic and number theoretic interpretations
of covering results seem to have attracted less attention than corresponding
interpretations of packing results. Yet, in the last two or three decades, covering
with convex bodies has become important in the local theory of normed spaces, see
the report of Giannopoulos and Milman [375], and Schneider [905] discovered that