Gruber P. Convex and Discrete Geometry

238 Convex Bodies
geometry, see Seidel [924] and, in particular, the monograph of Groemer [405]. The
Proceedings on Fourier analysis and convexity [343] and the book of Koldobsky
[606] also contain many results in convexity and the geometry of numbers based on
Fourier series, Fourier transforms, and spherical harmonics.
In the following we give the definitions and properties that will be used below.
A polynomial h : E d → R is harmonic if it satisfies the Laplace equation
�h = ∂2 h
∂x 2 1
+···+ ∂2 h
∂x 2 d
= 0.
The restriction of a homogeneous harmonic polynomial h : Ed → R to the unit
sphere Sd−1 is a spherical harmonic.Forn = 0, 1,...,let Hd n be the linear space of
all spherical harmonics in d variables of degree n and let
H d = H d 0 ⊕ Hd 1 ⊕ Hd 2 ⊕···
be the linear space of all finite sums of spherical harmonics in d variables. Let σ
denote the ordinary surface area measure in E d .Byarotation in E d an orthogonal
transformation with determinant 1 is meant and κd = V (B d ).
Proposition 13.1. We have the following statements:
(i) dim Hd � �
2n + d − 2 n + d − 2
n = .
n + d − 2 d − 2
�
(ii) 〈h, k〉 = h(u)k(u) dσ(u) = 0 for h ∈ Hd m , k ∈ Hd n , m �= n.
S d−1
(iii) The spherical harmonics in Hd 1 are the functions of the form
u → a · uforu∈ S d−1 , where a ∈ E d .
For the norm �·�2 on Hd n related to the inner product 〈·, ·〉, we have
�ui�2 = κ 1 2
d for u = (u1,...,ud) ∈ S d−1 .
(iv) Let h ∈ Hd n and r : Ed → Ed a rotation. Then rh ∈ Hd n , where rh is defined
by rh(u) = h � r −1 (u) � for u ∈ Sd−1 .
(v) If H is a linear subspace of Hd n that is invariant under rotations, then H ={0}
or Hd n .
(vi) Let h ∈ Hd . Then h + α is (the restriction to Sd−1 of) the support function of
a suitable convex body if α>0 is sufficiently large.
(vii) The family of all convex bodies the support functions of which are finite sums
of spherical harmonics, i.e. are in Hd , is dense in C.