Gruber P. Convex and Discrete Geometry

240 Convex Bodies
We distinguish three cases:
n = 0: Since by the assumption of (1) we have, H(h0) = H(rh0) = rH(h0)
for any rotation r, the point H(h0) ∈ Ed is rotation invariant and thus is equal to o,
concluding the proof of (2) for n = 0.
n > 1: Then
dim H d n
= 2n + d − 2
n + d − 2
� �
n + d − 2
≥ 2n + d − 2 > d
d − 2
by Proposition 13.1(i). Consider the linear subspace H ={h ∈ Hd n : H(h) = o} of
Hd n . By the assumption of (1), rH = H for each rotation r. Hence H ={0} or Hd n
by Proposition 13.1(v). Since H is the kernel of the linear mapping H : Hd n → Ed
and dim Hd n > d, it follows that dim H > 0 and thus H = Hd n or, equivalently,
H(h) = o for each h ∈ Hd n . The proof of (2) for n > 1 is complete.
n = 1: By Proposition 13.1(iii), every h ∈ Hd 1 is of the form h(u) = a · u for
u ∈ Ed where a ∈ Ed . Thus one can define a linear transformation l : Ed → Ed by
l(a) = H(a · u) ∈ E d for a ∈ E d .
Using the assumption in (1), it then follows that
(3) (lr)(a) = l � r(a) � = H � (ra) · u � = H � a · (r −1 u) � = H � r(a · u) �
= rH(a · u) = r � l(a) � = (rl)(a) for a ∈ E d and all rotations r,
where in the expression r(a · u) it is assumed that r operates on u – note that a · u ∈
H d 1 . Hence the linear transformation l : Ed → E d commutes with each rotation
r : E d → E d . This will be used to show that
(4) l(a) = γ a for a ∈ E d , where γ is a suitable constant.
If every a ∈ Ed is an eigenvector of the linear transformation l : Ed → Ed , then all
eigenvalues are the same and Proposition (4) holds. Otherwise choose an a ∈ Ed such that l(a) is not a multiple of a. Letrbe a rotation such that ra = a but
rl(a) �= l(a). Then rl(a) = lr(a) = l(a) by (3). This contradiction concludes
the proof of (4). We now show that (2) holds for h1 where h1(u) = a · u according to
Proposition 13.1(iii). Noting the definition of l(·), (4) and since for the Steiner point
of the convex body {a} we have s{a} = a, it then follows that
H(h1) = H(a · u) = l(a) = γ a = γ s{a} = γ
�
h{a}(u)udσ(u)
= γ
�
κd
Sd−1 (a · u)udσ(u) = γ
�
κk
Sd−1 κd
Sd−1 h1(u)udσ(u),
where h{a}(u) = a · u is the support function of the convex body {a}. This proves (2)
for n = 1 where β = γ/κd. The proof of (2) and thus of (1) is complete.
In the second step a particular linear map H : H d → E d is constructed which
satisfies the assumptions of (1). Let s : C → E d satisfy (ii). Then
(5) s is positive homogeneous of degree 1.