Gruber P. Convex and Discrete Geometry

242 Convex Bodies
by Proposition 13.1(iii). Thus β = 1/κd and (9) shows that
s(C) = 1
�
hC(u)udσ(u) = sC,
κd
Sd−1 concluding the proof of (8).
In the last step note that s(·) and sC are continuous in C. Hence (8) implies that
s(C) = sC for all C ∈ C, i.e. Proposition (ii) holds. ⊓⊔
Remark. For a slight refinement of this result, see Posicel’skiĭ [814]. The problem
remains whether one can relax the properties (b) and (c).
Description of the Continuous Endomorphisms of the Convex Cone 〈C, +〉
The next result is due to Schneider [900].
Theorem 13.7. Let d ≥ 3 and let E : C → C be a mapping. Then the following
statements are equivalent:
(i) E(C) = C + λ(C − C) for C ∈ C, where λ ≥ 0 is a constant.
(ii) E has the following properties:
(a) E(C + D) = E(C) + E(D) for C, D ∈ C
(b) E(aC) = aE(C) for C ∈ C and all surjective affinities a : E d → E d
(c) E is continuous
Description of the Endomorphisms of the Lattice 〈C, ∧, ∨〉
By an endomorphism of the lattice 〈C, ∧, ∨〉 a mapping E : C → C is meant for
which
E(C ∧ D) = E(C) ∧ E(D), E(C ∨ D) = E(C) ∨ E(D) for C, D ∈ C.
As a final result of this section we state a description by Gruber [424] of the endomorphisms
of 〈C, ∧, ∨〉.
Theorem 13.8. Let d ≥ 2 and let E : C → C be a mapping. Then the following
statements are equivalent:
(i) E is an endomorphism of 〈C, ∧, ∨〉.
(ii) For E one of the following hold:
(a) E(C) = DforC∈ C, where D is a fixed convex body
(b) There is a surjective affinity a : E d → E d such that E(C) = aC for C ∈ C
Remark. For a description of the endomorphisms in case d = 1, see [424].