Hadwiger's theorem

In integral geometry, Hadwiger's theorem characterises the valuations on convex bodies in It was proved by Hugo Hadwiger.

Introduction

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

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,

Any continuous valuation on that is invariant under rigid motions can be represented as

Any continuous valuation on that is invariant under rigid motions and homogeneous of degree is a multiple of