In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in \R^n. It was proved by Hugo Hadwiger.

Introduction

Valuations

Let \mathbb{K}^n be the collection of all compact convex sets in \R^n. A valuation is a function v : \mathbb{K}^n \to \R such that v(\varnothing) = 0 and for every S, T \in \mathbb{K}^n that satisfy S \cup T \in \mathbb{K}^n,

v(S) + v(T) = v(S \cap T) + v(S \cup T)~.

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(\varphi(S)) = v(S) whenever S \in \mathbb{K}^n and \varphi is either a translation or a rotation of \R^n.

Quermassintegrals

Main article: quermassintegral

The quermassintegrals W_j : \mathbb{K}^n \to \R are defined via Steiner's formula

\mathrm{Vol}_n(K + t B) = \sum_{j=0}^n \binom{n}{j} W_j(K) t^j~,

where B is the Euclidean ball. For example, W_0 is the volume, W_1 is proportional to the surface measure, W_{n-1} is proportional to the mean width, and W_n is the constant \operatorname{Vol}_n(B).

W_j is a valuation which is homogeneous of degree n - j, that is,

W_j(tK) = t^{n-j} W_j(K)~, \quad t \geq 0~.

Statement

Any continuous valuation v on \mathbb{K}^n that is invariant under rigid motions can be represented as

v(S) = \sum_{j=0}^n c_j W_j(S)~.

Corollary

Any continuous valuation v on \mathbb{K}^n that is invariant under rigid motions and homogeneous of degree j is a multiple of W_{n-j}.

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