Gruber P. Convex and Discrete Geometry

380 Geometry of Numbers
Hence li · m j �= 0 for suitable i, j, where 1 ≤ i ≤ d − k + 1, and 1 ≤ j ≤ k. Then,
Propositions (1) and (10) yield the left inequality:
λd−k+1λ ∗ k ≥ λiλ ∗ j
≥ 1.
We now prove the right-hand inequality. The second fundamental theorem shows
that
λ1 ···λd V (C) ≤ 2 d d(L), λ ∗ 1 ···λ∗ d V (C∗ ) ≤ 2 d d(L ∗ ).
Thus
(λ1λ ∗ d ) ···(λd−k+1λ ∗ k ) ···(λdλ ∗ 1 )V (C)V (C∗ ) ≤ 4 d d(L)d(L ∗ ).
Combining this and the inequalities
1 ≤ λ1λ ∗ d ,...,λd−k+1λ ∗ k ,...,λdλ ∗ 1 ,
which follow from the left inequality in (7), Propositions (8) and (9), we obtain the
right inequality. ⊓⊔
Remark. The right inequality has been refined substantially. The case where C = B d
has attracted particular attention. For a discussion and references, see Gruber [430].
A Relation Between Successive Minima and the Roots of the Ehrhart
Polynomial
Given a proper lattice polytope P ∈ P Z d , Ehrhart’s polynomiality theorem for the
lattice point enumerator L shows that
L(nP) = #(nP ∩ Z d ) = pP(n), n ∈ N,
where pP is a polynomial of degree d, called the Ehrhart polynomial of P. See
Sect. 19.1. The result of Henk, Schürmann and Wills is as follows.
Theorem 23.3. Let P ∈ P Z d be a proper, o-symmetric lattice polytope in E d . Let
λi = λi(P, Z d ) be its successive minima with respect to the integer lattice Z d and
let −γi =−γi(P, Z d ) be the roots of its Ehrhart polynomial. Then
γ1 +···+γd ≤ 1
2 (λ1 +···+λd).
Equality is attained for the cube P ={x :−1 ≤ xi ≤ 1}.
23.2 Jarník’s Transference Theorem and a Theorem of Perron and Khintchine
Let C be an o-symmetric convex body and L a lattice in E d . The family {C + l : l ∈
L} of translates of C by the vectors of L is called a set lattice. If any two distinct
translates have disjoint interiors, the set lattice is a lattice packing of C with packing
lattice L. If the translates cover E d , the set lattice is a lattice covering of C with
covering lattice L.