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Hadwiger's theorem

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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.

Contents

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,

v ( S ) + v ( T ) = v ( S ∩ T ) + v ( S ∪ 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 Wj: Kn → R are defined via Steiner's formula

V o l n ( K + t B ) = ∑ j = 0 n ( 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, Wn-1 is proportional to the mean width, and Wn is the constant Voln(B).

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

W j ( t K ) = t n − j W j ( K )   , t ≥ 0   .

Statement

Any continuous valuation v on Kn that is invariant under rigid motions can be represented as

v ( S ) = ∑ 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 Wn-j.

References

Hadwiger's theorem Wikipedia
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