Gruber P. Convex and Discrete Geometry

31 Covering with Convex Bodies 461
by (1). Since the mean value extended over all choices of m points in F of the
proportion in question is (1 − 1/s d ) m , there is at least one choice X ={x1,...,xm}
of m points in F for which this proportion is at most (1 − 1/s d ) m , concluding the
proof of (4).
Second, we prove the following proposition where the set X is as in (4).
(5) Let 0 < t ≤ 1/d and Y ={y1,...,yn} ⊆F with n maximum such that
the family � − tC + l + y : l ∈ L, y ∈ Y � is a packing contained in
E d \(C + L + X). Then
n ≤ sd
td �
1 − 1
sd �m .
This is easy. First, the proportion of space covered by the packing � −tC+l + y : l ∈
L, y ∈ Y � , that is its density, is nt d V (C)/V (F) = nt d /s d by (1) and the choice of
L = s Z d . Second, this packing is contained in E d \(C + L + X) and the proportion
of space covered by the latter set is at most (1 − 1/s d ) m by (4). The proof of (5) is
complete.
Third, note (4) and (5). In order to prove that
(6)
� (1 + t)C + l + z : l ∈ L, z ∈ X ∪ Y � is a covering,
let w ∈ Ed . We have to show that w is contained in at least one of the bodies in (6).
By (2), (3) and since by (5) 0 < t ≤ 1 d , the bodies −tC+m+w, m ∈ L, are pairwise
disjoint. We distinguish two cases. First, (−tC + L + w) ∩ (C + L + X) �= ∅. Then
there are p, q ∈ C, l, m ∈ L, xi ∈ X such that −tp + l + w = q + m + xi, or
w = tp + q − l + m + xi ∈ (1 + t)C + L + X. Second, if the first alternative does
not hold, then −tC + L + w ⊆ Ed � � \(C + L + X). By the maximality of the family
− tC + l + y : l ∈ L, y ∈ Y we then have (−tC + L + w) ∩ (−tC + L + Y ) �= ∅.
Thus there are p, q ∈ C, l, m ∈ L, y j ∈ Y such that −tp+l + w =−td+ m + y j ,
or w = tp + tq − l + m + y j ∈ (1 + t)C + L + Y by (2) and since 0 < t ≤ 1/d
by (5). This concludes the proof of (6).
Fourth, noting (1), the definition of L = s Zd , Propositions (4) and (5) show that
(7) the density of the covering in (6) is
(1 + t) d (m + n)
s d
≤ (1 + t) d
�
m 1
+
sd td �
1 − 1
sd �m� .
By choosing s, m, t suitably this finally yields the desired upper bound for a covering
of E d by translates of (1 + t)C, and thus for a corresponding covering of E d by
translates of C:Let
m 1
= d log
sd t
, t =
1
d log d .