Gruber P. Convex and Discrete Geometry

492 Geometry of Numbers
Let Dni be the Dirichlet–Voronoĭ cell of xni in K with respect to the set {xn1,...,
xnn}. Then Dni ⊆ ρn Bd + xni and the cells Dni form a tiling of the unit cube K .
This yields the following rough estimate:
n α d I (K,α,n) ≤ n α �
d min{�x
− xni�
xni
K
α } dx
�
�
�x − xni� α dx ≤ n α �
d V (Dni)ρ α n = n α d ρ α n .
Thus,
= n α d
i
Dni
(27) n α d I (K,α,n) ≤ (2d log d) α d
V (B d ) α d
.
Since V (B d ) = π d 2 /Ɣ(1 + d 2 ), Stirling’s formula for the gamma function and
the inequalities (26) and (27) yield the asymptotic formula for δα,d, as required. ⊓⊔
33.3 Structure of Minimizing Configurations, and a Heuristic Principle
While it is out of reach to give a precise description of the configurations Sn which
minimize the expressions (24) and (25) for all n, information is available as n →∞.
For d = 2 the minimizing configurations Sn are asymptotically regular hexagonal
and for d ≥ 3, they are still distributed rather regularly over J.
Without giving details, we roughly outline what is known. For precise information,
see the author [439, 442, 443]. In addition, a conjecture of Gersho [371] on the
distribution of Sn over J will be discussed.
The Case d = 2
For a wide class of strictly increasing functions f :[0, +∞) →[0, +∞) a weak
stability result of Gruber [439] says the following: For J in E 2 or on a Riemannian
2-manifold 〈M,ρM,ωM〉 and w = const the minimizing configurations Sn for the
expressions in (24) for the Euclidean norm and for (25) are asymptotically regular
hexagonal patterns in J as n →∞.Ifw is not constant then a result of this
type still holds, but its formulation is slightly more complicated. A related weak
stability result for the Euclidean norm for a wider class of functions f is due to
Fejes Tóth [321].
The Case d ≥ 2
A rather precise description of the minimizing sets Sn is indicated by the following
conjecture of Gersho [371] which we state without giving precise definitions.
Conjecture 33.1. There is a convex polytope P with V (P) = 1 which tiles E d with
congruent copies such that the following holds: Let J ⊆ E d be a Jordan measurable
i