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

## Introduction

### Valuations

Let Kn be the collection of all compact convex sets in Rn. A valuation is a function v:Kn → R such that v(∅) = 0 and, for every S,T ∈Kn for which STKn,

${\displaystyle v(S)+v(T)=v(S\cap T)+v(S\cup T)~.}$

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(φ(S)) = v(S) whenever S ∈ Kn and φ is either a translation or a rotation of Rn.

### Quermassintegrals

The quermassintegrals WjKn → R are defined via Steiner's formula

${\displaystyle \mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}~,}$

where B is the Euclidean ball. For example, W0 is the volume, W1 is proportional to the surface measure, Wn-1 is proportional to the mean width, and Wn is the constant Voln(B).

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

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

## Statement

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

${\displaystyle v(S)=\sum _{j=0}^{n}c_{j}W_{j}(S)~.}$

### Corollary

Any continuous valuation v on Kn that is invariant under rigid motions and homogeneous of degree j is a multiple of Wn-j.

## References

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