Hadwiger's theorem

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

Introduction

Valuations

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

A valuation is called continuous if it is continuous with respect to the Hausdorff metric. A valuation is called invariant under rigid motions if ${\displaystyle v(\varphi (S))=v(S)}$ whenever ${\displaystyle S\in \mathbb {K} ^{n}}$ and ${\displaystyle \varphi }$ is either a translation or a rotation of ${\displaystyle \mathbb {R} ^{n}.}$

Quermassintegrals

Main article: quermassintegral

The quermassintegrals ${\displaystyle W_{j}:\mathbb {K} ^{n}\to \mathbb {R} }$ 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 ${\displaystyle B}$ is the Euclidean ball. For example, ${\displaystyle W_{0}}$ is the volume, ${\displaystyle W_{1}}$ is proportional to the surface measure, ${\displaystyle W_{n-1}}$ is proportional to the mean width, and ${\displaystyle W_{n}}$ is the constant ${\displaystyle \operatorname {Vol} _{n}(B).}$

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

Statement

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

See also

• Minkowski functional – Function made from a set
• Set function – Function from sets to numbers