Gruber P. Convex and Discrete Geometry

13 The Space of Convex Bodies 237
Remark. A more general local version of this result has been obtained by
Weisshaupt [1017].
13.4 On the Algebraic Structure of C
The space C of convex bodies is an Abelian semigroup with respect to Minkowski
addition + on which the non-negative reals operate. Since 〈C, +〉 satisfies the cancellation
law, it can be embedded into an Abelian group or, more precisely, into a vector
space over R. The embedding can be achieved by considering equivalence classes
of pairs of convex bodies (as in the construction of the integers from the natural
numbers) or via support functions. For some references to the large pertinent literature,
see [428]. A different line of research deals with the characterization of homomorphisms
with additional properties of 〈C, +〉 into itself, into 〈E d , +〉 and into
〈R, +〉. Major contributions of this type are due to Schneider.
The definitions
C ∧ D = C ∩ D, C ∨ D = conv(C ∪ D) for C, D ∈ C
make C into an atomic lattice 〈C, ∧, ∨〉. It was investigated mainly by Belgian mathematicians.
In addition, a characterization of the endomorphisms of 〈C, ∧, ∨〉 has been
given.
Linearity and lattice properties of C were used as axioms for so-called convexity
spaces with the aim to raise convex geometry to a more general level of abstraction.
Other attempts to define convexity spaces are based on combinatorial results such
as the theorems of Caratheódory, Helly and Radon. See Sect. 3.2 and the references
cited there.
In the following a result of Schneider [898] on homomorphisms of 〈C, +〉 into
〈E d , +〉 is presented first. It deals with the Steiner point or curvature centroid of convex
bodies. Besides the centroid, the centres of the inscribed ellipsoid of maximum
volume and the circumscribed ellipsoid of minimum volume and other points, the
curvature centroid is one of the points which are assigned in a natural way to a convex
body. Tools for Schneider’s proof are spherical harmonics. We state the needed
definitions and some properties of the latter. Finally, two results on endomorphisms
of 〈C, +〉 and 〈C, ∧, ∨〉 due to Schneider [900] and Gruber [424], respectively, are
given without proof.
For more information and references to the original literature, see the articles and
surveys of Schneider [899], McMullen and Schneider [716], Saint-Pierre [874] and
the author [428] and the books of Schneider [907], Sect. 3.4 and Groemer [405],
Sect. 5.8.
Spherical Harmonics
Spherical harmonics are important tools of analysis and, in particular, of pure and
applied potential theory. Their first applications to problems of convex geometry
date back to Hurwitz [532] and Minkowski [741]. For expositions in the spirit of