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 edit
Valuations edit
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 edit
The quermassintegrals are defined via Steiner's formula
is a valuation which is homogeneous of degree that is,
Statement edit
Any continuous valuation on that is invariant under rigid motions can be represented as
Corollary edit
Any continuous valuation on that is invariant under rigid motions and homogeneous of degree is a multiple of
See also edit
- Minkowski functional – Function made from a set
- Set function – Function from sets to numbers
References edit
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
- Chen, B. (2004). "A simplified elementary proof of Hadwiger's volume theorem". Geom. Dedicata. 105: 107–120. doi:10.1023/b:geom.0000024665.02286.46. MR 2057247.