Gruber P. Convex and Discrete Geometry

26 The Torus Group E d /L 399
theory, see Mann [687, 688] and Danilov [235], and in other areas of mathematics.
Minkowski’s theorem on mixed volumes 6.5 and the Brunn–Minkowski theorem 8.1
are also α + β type results.
The sum theorem of Macbeath and Kneser, which will be proved later, gives a
satisfying answer to this question for the torus group. Let L be a lattice in E d .
The Sum Theorem of Kneser–Macbeath
Macbeath [673] and Kneser [601, 602] proved the following result.
Theorem 26.1. Let S, T be non-empty sets in E d /L such that S, T and S + T are
measurable. Then the following claims hold:
(i) If m(S) + m(T) >d(L), then S + T = E d /L.
(ii) If m(S) + m(T) ≤ d(L), then m(S + T) ≥ m(S) + m(T).
Note that the Lebesgue measurability of S and T does not necessarily imply
the Lebesgue measurability of S + T. IfS and T are Borel, their sum is Lebesgue
measurable, but need not be Borel. For references, see the Remark in Sect. 8.2.
If S + T is not measurable, a result similar to the sum theorem of Kneser and
Macbeath holds, where m(S + T) is replaced by the corresponding inner measure of
S + T.
We give two proofs. The first one is due to Kneser [602], see also Cassels [195].
The second proof is due to Macbeath [673] and Kneser [601]. In the second proof,
we consider only the case where S, T and S + T all are Jordan measurable sets.
Proof of Kneser
As a preparation for the first proof, we show the following result; versions of it
are useful tools in integral geometry. See, e.g. Hadwiger [468], Santaló [881] or
Schneider and Weil [911].
Lemma 26.1. Let U, V ⊆ E d /L be measurable. Then there is z ∈ E d /L such that:
m � (U + z) ∩ V � d(L) = m(U) m(V).
Proof. Let 1U and 1V be the characteristic functions of U and V, respectively. Then
m � (U + y) ∩ V � =
=
�
E d /L
�
E d /L
1U+y(x)1V(x) dm(x)
1U(x − y)1V(x) dm(x).