Hadwiger's theorem
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.