Gruber P. Convex and Discrete Geometry

The Steiner Point
13 The Space of Convex Bodies 239
The Steiner point or curvature centroid sC of a convex body C is the point
sC = 1
�
hC(u)udσ(u),
κd
Sd−1 where the integral is to be understood componentwise. If C is a proper convex body
of class C 2 , it can be shown that sC is the centroid of mass distributed over bd C with
density equal to the Gauss curvature of bd C. A nice application of the Steiner point
to approximate matching of shapes was given by Aichholzer, Alt and Rote [5].
Homomorphisms of 〈C, +〉 into 〈E d , +〉 and a Characterization of the Steiner
Point
The following result was proved by Schneider [898].
Theorem 13.6. Let s : C → E d be a mapping. Then the following statements are
equivalent:
(i) s(C) = sC for C ∈ C.
(ii) s satisfies the properties:
(a) s(C + D) = s(C) + s(D) for C, D ∈ C
(b) s(mC) = ms(C) for C ∈ C and all rigid motions m : E d → E d
(c) s(·) is continuous
Proof. (i)⇒(ii) This is a simple exercise.
(ii)⇒(i) The first step is to show the following:
(1) Let H : Hd → Ed be a linear mapping such that H(rh) = rH(h) for
h ∈ Hd and each rotation r : Ed → Ed . Then
�
H(h) = β h(u)udσ(u) for h ∈ H d ,
S d−1
where β is a real constant depending only on d and H and the integral is
considered componentwise.
To prove this, let h = h0 + h1 +···+hm where hn ∈ Hd n . From Proposition 13.1(ii)
and the fact that each coordinate ui of u ∈ Sd−1 is a spherical harmonic of degree 1
in Hd 1 by Proposition 13.1(iii), it follows that
�
�
h(u)udσ(u) = h1(u)udσ(u).
S d−1
S d−1
For the proof of (1) it is thus sufficient to show that
�
(2) H(hn) = β h1(u)udσ(u) for n = 1 and = o for n �= 1.
S d−1