Gruber P. Convex and Discrete Geometry

426 Geometry of Numbers
where S(·) stands for ordinary surface area in E d . Thus,
1
(2τ) d
�
V � (B d + t) ∩ τ K � ≤
t∈T
d + 2
2
2 − d 2
� √ �d 2(τ + 2 2)
.
(2τ) d
Now let τ →∞to get Proposition (3).
Having proved (3), the definition of δT (B d ) readily yields the theorem. ⊓⊔
More Precise Upper Estimates for δT (B d )
A slight improvement of Blichfeldt’s result is due to Rogers [849]:
δT (B d ) ≤ (d/e)2 −0.5d (1 + o(1)).
A refinement of Rogers’ upper estimate for small dimensions is due to Bezdek [111].
For many decades, it was a widespread belief among number theorists that, in
essence, Blichfeldt’s upper estimate for δT (B d ) was best possible. Thus it was a great
surprise when, in the 1970s, essential improvements were achieved using spherical
harmonics:
δT (B d ) ≤ 2 −0.5096d+o(d) : Sidel’nikov [934]
δT (B d ) ≤ 2 −0.5237d+o(d) :Levenˇstein [651]
δT (B d ) ≤ 2 −0.599d+o(d) : Kabat’janski and Levenˇstein [557]
For an outline of the proof of the last estimate, see Fejes Tóth and Kuperberg [325].
Lower Estimate for δL(B d )
In Sect. 30.3 we shall see, in the more general context of lattice packing of convex
bodies, that, as a consequence of the Minkowski–Hlawka theorem 24.1, the following
result holds.
Theorem 29.3. δL(B d ) ≥ 2 −d .
Actually, slightly more is true, where ζ(·) denotes the Riemann zeta function:
δL(B d ) ≥ 2ζ(d) 2 −d . This follows from Hlawka’s [509] version of the
Minkowski–Hlawka theorem for star bodies.
δL(B d ) ≥ 2dcd 2 −d , where cd → log √ 2asd →∞. This follows from
Schmidt’s [893] refinement of the Minkowski–Hlawka theorem.
δL(B d ) ≥ 2(d − 1)ζ(d) 2 −d . This is the best known lower bound. It is
due to Ball [52].
It is believed that no essential improvement of these estimates is possible in the sense
that the best estimate is of the form
δL(B d ) ≥ 2 −d+o(d) as d →∞.