Hadwiger's theorem

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

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 $S \in \mathbb^n$ and $\varphi$ is either a translation or a rotation of $\R^n.$

Quermassintegrals

The quermassintegrals $W_j : \mathbb^n \to \R$ are defined via Steiner's formula
$\mathrm_n(K + t B) = \sum_^n \binom W_j(K) t^j~,$
where $B$ is the Euclidean ball. For example, $W_0$ is the volume, $W_1$ is proportional to the surface measure, $W_$ is proportional to the mean width, and $W_n$ is the constant $\operatorname_n(B).$

$W_j$ is a valuation which is homogeneous of degree $n - j,$ that is,
$W_j(tK) = t^ W_j(K)~, \quad t \geq 0~.$

Statement

Any continuous valuation $v$ on $\mathbb^n$ that is invariant under rigid motions can be represented as
$v(S) = \sum_^n c_j W_j(S)~.$

Corollary

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

See also

*
*

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
* {{cite journal, title=A simplified elementary proof of Hadwiger's volume theorem, journal=Geom. Dedicata, volume=105, year=2004, pages=107–120, last=Chen, first=B., mr=2057247, doi=10.1023/b:geom.0000024665.02286.46

Integral geometry
Theorems in convex geometry
Theorems in probability theory