Hadwiger's theorem

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

Introduction

Valuations

Let $\mathbb {K} ^{n}$ be the collection of all compact convex sets in $\mathbb {R} ^{n}.$ A valuation is a function $v:\mathbb {K} ^{n}\to \mathbb {R}$ 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 $\mathbb {R} ^{n}.$

Quermassintegrals

The quermassintegrals $W_{j}:\mathbb {K} ^{n}\to \mathbb {R}$ are defined via Steiner's formula $\mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}~,$ where $B$ is the Euclidean ball. For example, $W_{0}$ 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

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.

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