In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in
It was proved by Hugo Hadwiger.
be the collection of all compact convex sets in
A valuation is a function
that satisfy
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 (
φ
is either a translation or a rotation of
The quermassintegrals
are defined via Steiner's formula
{\displaystyle \mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}~,}
is the Euclidean ball.
is the volume,
is proportional to the surface measure,
is proportional to the mean width, and
is the constant
Vol
{\displaystyle \operatorname {Vol} _{n}(B).}
is a valuation which is homogeneous of degree
{\displaystyle n-j,}
{\displaystyle W_{j}(tK)=t^{n-j}W_{j}(K)~,\quad t\geq 0~.}
Any continuous valuation
that is invariant under rigid motions can be represented as
Any continuous valuation
that is invariant under rigid motions and homogeneous of degree
is a multiple of
{\displaystyle W_{n-j}.}
An account and a proof of Hadwiger's theorem may be found in An elementary and self-contained proof was given by Beifang Chen in