Gruber P. Convex and Discrete Geometry

A Lower Estimate for the Maximum Lattice Packing Density
30 Packing of Convex Bodies 449
As a consequence of the Minkowski-Hlawka theorem and the inequality of Rogers
and Shephard on difference bodies, we obtain the following estimates:
Theorem 30.4. Let C be a proper convex body in E d . Then, as d →∞,
(i) δL(C) ≥ 2 −d if C is centrally symmetric.
(ii) δL(C) ≥ 4 −d+o(d) for general C.
Proof. (i) Let C be an o-symmetric proper convex body. The Minkowski–Hlawka
theorem 24.1 then shows that there are lattices L which contain no point of 2C,
except o and with determinant d(L) greater than but arbitrarily close to V (2C). These
lattices provide packings of C where the densities V (C)/d(L) are less than, but
arbitrarily close to, 2−d . This clearly yields δL(C) ≥ 2−d .
(ii) Let C be a proper convex body. Its central symmetrization D = 1 2 (C − C)
then is a proper, o-symmetric convex body. By (i), there are lattices L which provide
packings of D with density V (D)/d(L) arbitrarily close to 2−d . By Proposition 30.3,
each such lattice provides a packing of C where for the density we have
V (C)
d(L)
V (C) V (D)
=
V (D) d(L)
≥ 2d
� 2d
d
�
V (D)
d(L)
by the Rogers–Shephard inequality. Since this is arbitrarily close to
2d �2 −d = 4 −d+o(d) ,
� 2d
d
we obtain δL(C) ≥ 4 −d+o(d) . ⊓⊔
Heuristic Observations
We now extend the heuristic observations in Sect. 24.2. There is reason to believe
that the bound in (i) cannot be improved essentially for certain o-symmetric convex
bodies, perhaps even for Euclidean balls. If this is true, there is a function ψ : N → R
where
ψ(d) = o(d), ψ(d) →∞as d →∞,
such that the following hold: for each d there is an o-symmetric convex body C
in E d with V (2C) = 2 −ψ(d) and each lattice L ∈ L(1) contains a point �= o in
the interior of the convex body 2 1+2ψ(d)/d C of volume 2 ψ(d) . Thus, no lattice L ∈
L(1) provides a packing of the convex body 2 2ψ(d)/d C of volume 2 −d+ψ(d) .This
implies that δL(2 2ψ(d)/d C) ≤ 2 −d+ψ(d) . Since the maximum lattice packing density
is invariant with respect to dilatations we have,
δL(C) ≤ 2 −d+ψ(d) .