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 $\mathbb{K}^n$ 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 =

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

{{cite book|mr=1608265|last=Klain|first=D.A.|last2=Rota|author2-link=Gian-Carlo Rota|first2=G.-C.|title=Introduction to geometric probability|url=https://archive.org/details/introductiontoge0000klai|url-access=registration|publisher=Cambridge University Press|location=Cambridge|year=1997|isbn=0-521-59362-X}}

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}}

Category:Integral geometry

Category:Theorems in convex geometry

Category:Theorems in probability theory