Gruber P. Convex and Discrete Geometry

n=1
13 The Space of Convex Bodies 235
Here,
� �
∞�
∞�
µh(A) = lim inf h(diam Un) : Un ⊆ D, diam Un ≤ ε, A ⊆
ε→+0
where diam is the diameter with respect to the given metric δ.
In this section we prove the result of Bandt and Baraki.
Non-Existence of Isometry-Invariant Measures on 〈C,δ H 〉
A Borel measure µ on C is isometry-invariant with respect to δ H if
µ(D) = µ � I (D) � for all Borel sets D ⊆ C and each isometry I : C → C
with respect to δ H .
These isometries have been determined by Gruber and Lettl [449], see Theorem 13.4
below. This result shows that there are few isometries of 〈C,δ H 〉 into itself. Thus the
condition that a measure µ on C is isometry-invariant is not too restrictive. In spite
of this we have the following negative result of Bandt and Baraki [66].
Theorem 13.3. Let d > 1. Then there is no positive σ -finite Borel measure on C
which is invariant with respect to all isometries of 〈C,δ H 〉 into itself.
Proof. Assume that there is a positive σ -finite Borel measure µ on C which is
isometry-invariant with respect to δ H .Let
Cn = C(nB d ) ={C ∈ C : C ⊆ nB d } for n = 1, 2,...
Since C is the union of the compact sets Cn and µ is positive, there is an n with
(1) µ(Cn) >0.
For this n we have the following:
(2) C contains uncountably many pairwise disjoint
isometric copies of Cn.
To see this, we first show that
(3) the sets Cn +[o, p] ={C +[o, p] :C ∈ Cn}, p ∈ 3nS d−1 ,
are pairwise disjoint.
If (3) did not hold, there are p, q ∈ 3nS d−1 , p �= q, such that C+[o, p] =D+[o, q]
for suitable C, D ∈ Cn. Choose a linear form l on E d such that l(p) = 1, l(q) = 0.
Let r ∈ C be such that
l(r) = min{l(x) : x ∈ C}.
Choose s ∈ D and 0 ≤ λ ≤ 1, such that r = s + λq. Choose t ∈ C and 0 ≤ ν ≤ 1,
such that s + q = t + νp. Then t + νp − q + λq = r and thus
l(t) + ν = l(t + νp − q + λq) = l(r) ≤ l(t).
n=1
Un
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