Gruber P. Convex and Discrete Geometry

222 Convex Bodies
To show that (K + x) ∩ (K + y) ⊆ K + x ∨ y, letw ∈ (K + x) ∩ (K + y).
Then w ∈ K + x and w ∈ K + y and thus x, y � w and therefore x ∨ y � w, or
w ∈ K + x ∨ y. To show the reverse inclusion, let w ∈ K + x ∨ y. Then x ∨ y � w
and thus x, y � w. This yields w ∈ (K + x) ∩ (K + y). The proof of (4) is complete.
Choose a hyperplane H as in the remarks before the theorem and let C = H ∩ K .
Then the following statement holds:
(5) Let λ, µ ≥ 0 and x, y ∈ E d with (λC + x) ∩ (µC + y) �= ∅. Then
(λC + x) ∩ (µC + y) = νC + x ∨ y for suitable ν ≥ 0.
Let G be the hyperplane parallel to H which contains λC + x and µC + y. Then
and thus,
λC + x = (K + x) ∩ G, µC + y = (K + y) ∩ G
(λC + x) ∩ (µC + y) = (K + x) ∩ (K + y) ∩ G
mm = (K + x ∨ y) ∩ G = νC + x ∨ yfor suitableν ≥ 0
by (4). This concludes the proof of (5).
(5) Together with Theorem 12.2 of Choquet and Rogers and Shephard implies
that C is a simplex. Hence K is a simplicial cone. To see that dim K = d, consider
a basis {b1,...,bd} of E d and note that for z = o ∧ b1 ∧···∧bd the cone K + z
contains o, b1,...,bd. Thus dim K = dim(K + z) = d.
(ii)⇒(i) This is easy to prove on noting that after a suitable linear transformation
we may assume that K ={x : 0 ≤ xi}. ⊓⊔
Remark. Choquet actually proved his result in infinite dimensions using Choquet
simplices instead of (finite dimensional) simplices. A generalization to vector spaces
without any topology is due to Kendall [574]. See the surveys by Peressini [789] and
Rosenthal [857].
12.2 A Characterization of Balls by Their Gravitational Fields
During the twentieth century a multitude of different characterizations of (Euclidean)
balls and spheres have been given. Besides elementary characterizations, characterizations
by extremal properties, in particular by properties of isoperimetric type,
and other characterizations in the context of convexity, there is a voluminous body
of differential geometric characterizations. Interesting sporadic characterizations of
balls have their origin in other branches of mathematics, for example in potential
theory and, even outside of mathematics.
In the sequel we consider a characterization of balls by their Newtonian gravitational
fields. We consider the case d = 3, but the result can be extended easily to any
d ≥ 2, where, for d = 2, logarithmic potentials have to be used.
Surveys of characterizations of balls are due to Bonnesen and Fenchel [149],
Giering [377], Burago and Zalgaller [178], Bigalke [114] and Heil and Martini [488].
We add two references, one related to cartography by Gruber [426] and one to
electrostatics by Mendez and Reichel [719]. For a characterization of balls using
topological tools see Montejano [750].