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

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.