Hadwiger's theorem

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

Introduction

Valuations

Let [math]\displaystyle{ \mathbb{K}^n }[/math] be the collection of all compact convex sets in [math]\displaystyle{ \R^n. }[/math] A valuation is a function [math]\displaystyle{ v : \mathbb{K}^n \to \R }[/math] such that [math]\displaystyle{ v(\varnothing) = 0 }[/math] and for every [math]\displaystyle{ S, T \in \mathbb{K}^n }[/math] that satisfy [math]\displaystyle{ S \cup T \in \mathbb{K}^n, }[/math] [math]\displaystyle{ 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]\displaystyle{ v(\varphi(S)) = v(S) }[/math] whenever [math]\displaystyle{ S \in \mathbb{K}^n }[/math] and [math]\displaystyle{ \varphi }[/math] is either a translation or a rotation of [math]\displaystyle{ \R^n. }[/math]

Quermassintegrals

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

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

Statement

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

Corollary

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

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.