Gruber P. Convex and Discrete Geometry

7 Valuations 133
Theorem 7.9. Let φ be a continuous valuation on C which is invariant with respect
to proper rigid motions. Then
φ = c0W0 +···+cd Wd with suitable constants c0,...,cd.
This result clearly shows the importance of the quermassintegrals W0,...,Wd on C.
For information on quermassintegrals, see Sect. 6.4 and the references cited there.
Proof (by Induction on d). The functional theorem is trivial for d = 0. Assume now
that d > 0 and that it holds for d − 1. We have to prove it for d. The restriction of
φ to C(E d−1 ) satisfies the assumption of the theorem. The induction hypothesis then
shows that
(18) φ(C) = d0w0(C) +···+dd−1wd−1(C) for C ∈ C(E d−1 ),
where d0,...,dd−1 are suitable constants and w0,...,wd−1 are the quermassintegrals
in E d−1 . Note that
d κi−1
(19) W0(C) = 0,wi−1(C) = Wi(C)
i κi
for C ∈ C(Ed−1 ) ⊆ C(Ed = Ed−1 × R),
by Proposition 6.7. Consider the valuation
ψ = φ −
d�
i=1
di−1
d κi−1
Wi = φ −
i κi
d�
ci Wi, say.
ψ(C) = 0forC∈ C(Ed−1 ) ⊆ C(Ed ) by (18) and (19). Since φ is invariant with
respect to proper rigid motions, by assumption, and the same is true of the quermassintegrals
Wi, by Theorem 6.13, ψ is also invariant with respect to proper rigid
motions. This shows that ψ is simple. Since φ is continuous, by assumption, and the
quermassintegrals Wi are continuous by Theorem 6.13, the valuation ψ is continuous
too. Thus ψ satisfies the assumptions of Hadwiger’s characterization theorem for the
volume which then implies that ψ = c0V = c0W0 where c0 is a suitable constant.
Hence
d�
d�
φ = ψ + ci Wi = ci Wi. ⊓⊔
i=1
Remark. Hadwiger [462, 464] also proved a similar result, where, instead of continuity,
monotony is assumed. An interesting recent result of Ludwig and Reitzner
[666] shows that the valuations on C which are semi-continuous and invariant with
respect to volume preserving affinities are precisely the linear combinations of the
Euler characteristic, affine surface area and volume. Alesker [7, 8] determined the
continuous rotation invariant and the continuous translation invariant valuations
on C. For further pertinent results and additional information, see McMullen and
Schneider [716], McMullen [714] and Klain and Rota [587].
i=0
i=1