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 $v(\phi (S))=v(S)$ whenever $S\in {\mathbb{𝕂}}^n$ and $\phi$ 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 $${\mathrm{V}\mathrm{o}\mathrm{l}}_n(K+tB)=\sum _{j=0}^n(\genfrac{}{}{0ex}{}{n}{j})W_j(K)t^j,$$ 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 ${\mathrm{Vol}}_n(B).$

$W_j$ is a valuation which is homogeneous of degree $n-j,$ that is, $$W_j(tK)=t^{n-j}{W}_j(K),t\ge 0.$$

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 $W_{n-j}.$

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. https://archive.org/details/introductiontoge0000klai. 

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.