Hadwiger's theorem

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

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 $S\in {𝕂}^n$ and $\phi$ is either a translation or a rotation of ${ℝ}^n.$

Quermassintegrals

The quermassintegrals $W_j:{𝕂}^n\to ℝ$ 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 ${𝕂}^n$ 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 ${𝕂}^n$ that is invariant under rigid motions and homogeneous of degree $j$ is a multiple of $W_{n-j}.$

See also

• Template:Annotated link
• Template:Annotated link

References

An account and a proof of Hadwiger's theorem may be found in

• Template:Cite book

An elementary and self-contained proof was given by Beifang Chen in

• Template:Cite journal