Gruber P. Convex and Discrete Geometry

404 Geometry of Numbers
Proof. Put ϱ = ϱ(C, L) and µ = µ(C, L), and let F be a fundamental parallelotope
of L.BytC for t ≥ 0wemeanthesettC + L. It follows from Proposition 26.4 that
(3) m(tC) = V (F ∩ tC) for t ≥ 0.
Using (3), we prove the following two assertions.
First,
(4) m(tC) = t d V (C) for 0 ≤ t ≤ ϱ.
For 0 ≤ t ≤ ϱ the set lattice {tC + l : l ∈ L} is a packing. Hence its density
V (tC)/d(L) = t d V (C)/d(L) is the proportion of E d covered by the sets tC +l,see
Sect. 30.1. The latter equals the proportion of F covered by the sets tC + l and thus
is equal to V � F ∩ (tC + L) � /d(L) = V (F ∩ tC)/d(L). Hence (3) yields (4).
Second,
(5) µ = inf � t > 0 : m(tC) = d(L) � .
If t > µ, then {tC + l : l ∈ L} is a covering. Then tC = tC + L = E d and
thus m(tC) = V (F) = d(L) by (3). If t < µ, then {tC + l : l ∈ L} is not a
covering. In particular, the family tC + l : l ∈ L of translates of the (compact)
convex body by vectors of the lattice L do not cover F. Hence m(tC) = V (F ∩tC) =
V � F ∩ (tC + L) � < V (F) = d(L) by (3). The proof of (5) is now complete.
Noting that (s + t)C ⊇ sC + tC, the sum theorem shows that
(6) m � (s + t)C � ≥ min � m(sC) + m(tC), d(L) � for s, t ≥ 0.
Having shown (4)–(6), the proof of the theorem is easy: By (4),
(7) m(ϱC) = ϱ d V (C), m(r 1 d ϱC) = rϱ d V (C).
Applying (6) several times, (7) and (1), it follows that
m � ϱ(q + r 1 d )C � ≥ min � qm(ϱC) + m(r 1 d ϱC), d(L) �
= min � ϱ d (q + r)V (C), d(L) � = d(L).
This, together with (5), finally yields (2). ⊓⊔
27 Special Problems in the Geometry of Numbers
In this chapter we have so far outlined systematic features of the geometry of
numbers. What about problems? In contrast to other areas of mathematics, in the
geometry of numbers there is only a rather small number of basic particular problems.
Roughly speaking, these are of two types. First, special problems of a more
arithmetic character, including problems on forms of various types. A selection of
these will be considered here. Second, sets of problems involving reduction, packing,
covering and tiling. While reduction has arithmetic and geometric aspects, the
other three sets of problems of the second group definitely are geometric. Reduction,
packing, covering and tiling will be studied in the subsequent sections. In this section
we consider the following special problems: