14 I. Convex Sets at Large
Hence we have pc ∈ H2k,n. Let
K = conv pc : c ∈
be the convex hull of all polynomials pc. Since the sphere
is compact and the
map c −→ pc is continuous, the set pc : c ∈ Sn−1 is a compact subset of H2k,n.
Therefore, by Corollary 2.4, we conclude that K is compact.
Let us prove that γ x 2k ∈ K for some γ 0. The idea is to average the
polynomials pc over all possible vectors c ∈ Sn−1. To this end, let dc be the
rotation invariant probability measure on Sn−1 and let
(3.3.1) p(x) =
pc(x) dc =
be the average of all polynomials pc. We observe that p ∈ H2k,n. Moreover, since dc
is a rotation invariant measure, we have U(p) = p for any orthogonal transformation
and hence by Lemma 3.2, we must have
p(x) = γ x
for some γ ∈ R.
We observe that γ 0. Indeed, for any x = 0, we have pc(x) 0 for all c ∈
except from a set of measure 0 and hence p(x) 0.
The integral (3.3.1) can be approximated with arbitrary precision by a finite
pci (x) for some ci ∈
Therefore, p lies in the closure of K. Since K is closed, p ∈ By 2.3, we
can write p(x) = γ x
as a convex combination of some
pci (x) = ci,x
Dividing by γ, we complete the proof.
It is not always easy to come up with a particular choice of ci in the identity
of Proposition 3.3.
1. Prove Liouville’s identity:
2. Prove Fleck’s identity:
ξi ±ξj ±ξk
where the sums containing ± signs are taken over all possible independent choices
of pluses and minuses.