Hadwiger's theorem

In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in Rⁿ. It was proved by Hugo Hadwiger.

Introduction

Valuations

Let Kⁿ be the collection of all compact convex sets in Rⁿ. A valuation is a function v:Kⁿ → R such that v(∅) = 0 and, for every S,T ∈Kⁿ for which S∪T∈Kⁿ,

${\displaystyle v(S)+v(T)=v(S\cap T)+v(S\cup T)~.}$

A valuation is called continuous if it is continuous with respect to the Hausdorff metric. A valuation is called invariant under rigid motions if v(φ(S)) = v(S) whenever S ∈ Kⁿ and φ is either a translation or a rotation of Rⁿ.

Quermassintegrals

The quermassintegrals W_j: Kⁿ → 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 B is the Euclidean ball. For example, W₀ is the volume, W₁ is proportional to the surface measure, W_n-1 is proportional to the mean width, and W_n is the constant Vol_n(B).

W_j is a valuation which is homogeneous of degree n-j, that is,

${\displaystyle W_{j}(tK)=t^{n-j}W_{j}(K)~,\quad t\geq 0~.}$

Statement

Any continuous valuation v on Kⁿ 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 v on Kⁿ that is invariant under rigid motions and homogeneous of degree j is a multiple of W_n-j.

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.