Hadwiger's theorem

Template:Short description In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in <math>\R^n.</math> It was proved by Hugo Hadwiger.

Introduction

Valuations

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

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

Quermassintegrals

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The quermassintegrals <math>W_j : \mathbb{K}^n \to \R</math> are defined via Steiner's formula <math display=block>\mathrm{Vol}_n(K + t B) = \sum_{j=0}^n \binom{n}{j} W_j(K) t^j~,</math> where <math>B</math> is the Euclidean ball. For example, <math>W_0</math> is the volume, <math>W_1</math> is proportional to the surface measure, <math>W_{n-1}</math> is proportional to the mean width, and <math>W_n</math> is the constant <math>\operatorname{Vol}_n(B).</math>

<math>W_j</math> is a valuation which is homogeneous of degree <math>n - j,</math> that is, <math display=block>W_j(tK) = t^{n-j} W_j(K)~, \quad t \geq 0~.</math>

Statement

Any continuous valuation <math>v</math> on <math>\mathbb{K}^n</math> that is invariant under rigid motions can be represented as <math display=block>v(S) = \sum_{j=0}^n c_j W_j(S)~.</math>

Corollary

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

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