Gruber P. Convex and Discrete Geometry

458 Geometry of Numbers
Proof. Call x, y ∈ Ed congruent modulo L if x − y ∈ L. The points of the form
1
2 l with l ∈ L then fall into 2d congruence classes modulo L. (To see this, let
{b1,...,bd} be a basis of L. Then each point of the form 1 2 l with l ∈ L is congruent
to one of the 2d points
u1
2 b1 +···+ ud
2 bd where ui ∈{0, 1}
and these points are pairwise incongruent.) Let o, 1 2 l1,..., 1 2 ln, with li ∈ L and
n = 2 d − 1 be representatives of these congruence classes. Then, as the translates of
C by the vectors of L cover E d , there are m1,...,mn ∈ L such that 1 2 li ∈ C + mi,
or
1
2 li − mi ∈ C for i = 1,...,n.
These points lie in different congruence classes modulo L and are not congruent
to o. Since −( 1 2 li − mi) ≡ 1 2 li − mi modulo L, we see that the points −( 1 2 li − mi)
and 1 2 li − mi lie in the same congruence class but clearly are different. Since C is
o-symmetric it follows that
the 2n = 2 d+1 − 2 points ±( 1 2 li − mi), i = 1,...,n,
are pairwise different, different from o and all are contained in C.
Thus C meets each of the 2n + 1 = 2 d+1 − 1 distinct bodies C, C ± (li − 2mi),
i = 1,...,n, of the covering. ⊓⊔
31.3 Rogers’s Upper Bound for ϑT (C)
Numerous early attempts to construct or to establish the existence of dense lattice
and non-lattice coverings of E d with convex bodies were relatively unsuccessful.
The upper bounds which were obtained all were of the form c d with suitable constants
c > 1. The breakthrough finally was achieved by Rogers who introduced
averaging methods to this problem which led to surprisingly small upper estimates
for the minimum density of a covering of E d by translates of a convex body and for
lattice coverings.
In this section we prove Rogers’ upper estimate for the minimum density of a
covering of E d by translates of a convex body. His proof is based on periodic sets
and a mean value argument.
For an exposition of the work of Rogers on coverings, the reader may wish to
consult Rogers’ classical Cambridge tract [851].
An Inequality Between δT (C) and ϑT (C)
Before embarking on Rogers’ result, we present a result which relates the packing
and the covering case and, as a consequence, yields a (large) upper bound for ϑT (C).
We point out the nice idea of proof, which might have been a starting point for
Rogers’ ingenious proof, see later.