Let Kn be the collection of all compact convex sets in Rn. A valuation is a function v:Kn → R such that v(∅) = 0 and, for every S,T ∈Kn for which S∪T∈Kn,
A valuation is called continuous if it is continuous with respect to the Hausdorff metric. A valuation is called invariant under rigid motions if v(φ(S)) = v(S) whenever S ∈ Kn and φ is either a translation or a rotation of Rn.
The quermassintegrals Wj: Kn → R are defined via Steiner's formula
Wj is a valuation which is homogeneous of degree n-j, that is,
Any continuous valuation v on Kn that is invariant under rigid motions can be represented as
Any continuous valuation v on Kn that is invariant under rigid motions and homogeneous of degree j is a multiple of Wn-j.
An account and a proof of Hadwiger's theorem may be found in
- Klain, D.A.; Rota, G.-C. (1997). Introduction to geometric probability. Cambridge: Cambridge University Press. ISBN 0-521-59362-X. MR 1608265.
An elementary and self-contained proof was given by Beifang Chen in