Hadwiger's theorem explained
In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in
It was proved by
Hugo Hadwiger.
Introduction
Valuations
Let
be the collection of all compact convex sets in
A
valuation is a function
such that
and for every
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
whenever
and
is either a
translation or a
rotation of
Quermassintegrals
The quermassintegrals
are defined via Steiner's formula
where
is the Euclidean ball. For example,
is the volume,
is proportional to the
surface measure,
is proportional to the
mean width, and
is the constant
is a valuation which is
homogeneous of degree
that is,
Statement
Any continuous valuation
on
that is invariant under rigid motions can be represented as
Corollary
Any continuous valuation
on
that is invariant under rigid motions and homogeneous of degree
is a multiple of
References
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
- A simplified elementary proof of Hadwiger's volume theorem. Geom. Dedicata. 105. 2004. 107 - 120. Chen. B.. 2057247. 10.1023/b:geom.0000024665.02286.46.