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} → **R** such that *v*(∅) = 0 and, for every *S*,*T* ∈**K**^{n} for which *S*∪*T*∈**K**^{n},

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

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* ∈ **K**^{n} and *φ* is either a translation or a rotation of **R**^{n}.

Quermassintegrals

Main article: quermassintegral

The quermassintegrals *W*_{j}: **K*** ^{n}* →

**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, *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 Vol_{n}(*B*).

*W*_{j} is a valuation which is homogeneous of degree *n*-*j*, that is,

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

Statement

Any continuous valuation v on K* ^{n}* 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 **K*** ^{n}* 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". Geometriae Dedicata. 105: 107–120. doi:10.1023/b:geom.0000024665.02286.46. MR 2057247.

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