Gruber P. Convex and Discrete Geometry

31 Covering with Convex Bodies 459
Proposition 31.1. Let C be a proper, o-symmetric convex body in E d . Then
(1 ≤) ϑT (C) ≤ 2 d δT (C)(≤ 2 d ). In particular, δT (C) ≥ 2 −d .
Proof. Consider a packing of translates of C in E d . By successively inserting additional
translates of C into the interstitial space of the packing, we finally arrive at
a packing of translates of C, say{C + t : t ∈ T }, such that, for any x ∈ E d ,the
translate C + x meets at least one translate of the packing, say C + t. Thus there are
p, q ∈ C such that p + x = q +t, and therefore x = q − p +t ∈ C −C +t = 2C +t
since C is o-symmetric. In other words, {2C + t : t ∈ T } is a covering of translates
of C. ⊓⊔
If C is an o-symmetric convex body for which the Minkowski–Hlawka bound
2 −d+o(d) for δL(C) is best possible, then ϑL(C) ≤ 2 o(d) . The upper estimates of
Rogers for ϑT (C) and ϑL(C) are more explicit and hold for all o-symmetric convex
bodies C.
Rogers’s Upper Bound for ϑT (C)
Rogers [848, 851] proved the following upper estimate for ϑT (C). See Füredi and
Kang [347] for an elegant proof of a slightly weaker result.
Theorem 31.4. Let C be a proper convex body in E d . Then
ϑT (C) ≤ d log d � 1 + o(1) � as d →∞.
The following rough outline of the proof may help the reader:
It is enough to cover a large box and to continue by periodicity.
Randomly placed translates of C cover almost the whole box
in an economic way.
With an additional trick the small holes are covered.
Proof. We may suppose that
(1) V (C) = 1
and that o is the centroid of C. Then a well-known result, which can easily be proved,
says that
(2) − 1
C ⊆ C.
d
Choose s > 0 so large that, for the lattice L = s Z d ,
(3) any two distinct bodies of the family {C + l : l ∈ L} are disjoint.
Let F ={x : 0 ≤ xi < s} be a fundamental parallelotope of L and let m be a large
positive integer.