Gruber P. Convex and Discrete Geometry

82 Convex Bodies
We are now ready to state some simple algebraic properties of the space C
of convex bodies in E d , endowed with the operations of Minkowski addition and
multiplication with positive numbers.
Theorem 6.1. The following claims hold:
(i) C, endowed with Minkowski addition, is a commutative semigroup with cancellation
law.
(ii) C, endowed with Minkowski addition, and multiplication with non-negative numbersisaconvex
cone, i.e. C + D, λC ∈ C for C, D ∈ C,λ≥ 0.
Proof. (i) Proposition 6.1 and the simple fact that Minkowski addition is associative
and commutative, settle the first part of statement (i). For the proof that the cancellation
law holds, let B, C, D ∈ C such that B + D = C + D. Then h B +h D = hC +h D
by Proposition 6.2. Therefore, h B = hC which, in turn, implies that B = C by
Proposition (4) before Theorem 4.3.
(ii) This is simply a re-statement of Proposition 6.1. ⊓⊔
Direct Sum Decomposition of Convex Bodies
A convex body C is the direct sum of the convex bodies C1,...,Cm,
if
C = C1 ⊕···⊕Cm,
C = C1 +···+Cm and lin C1 ⊕···⊕lin Cm exists.
By lin we mean the linear hull. The convex body C is directly irreducible if, in any
direct decomposition of C, at most one summand is different from {o}.
Our aim is to prove the following result of the author [412], II. For alternative
proofs and generalizations, see [412], Gale and Klee [352] and Kincses [584] and
Schneider [907]. The following proof is essentially that of Kincses.
Theorem 6.2. Let C ∈ Cp. Then there are directly irreducible convex bodies C1,...,
Cm ∈ C, such that
C = C1 ⊕···⊕Cm.
This decomposition is unique up to the order of the summands.
For the proof we need the following tool. For later reference we prove it in a slightly
more general form than is needed below. HC(u) is the support hyperplane of C with
exterior normal vector u.
Lemma 6.1. Let C1,...,Cm ∈ C, λ1,...,λm ≥ 0, and u ∈ S d−1 . Then, for C =
λ1C1 +···+λmCm, the following holds:
�
C ∩ HC(u) = λ1 C1 ∩ HC1 (u)� �
+···+λm Cm ∩ HCm (u)� .