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.

Introduction[edit]

Valuations[edit]

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 STKn,

 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[edit]

Main article: quermassintegral

The quermassintegrals WjKn → R are defined via Steiner's formula

 \mathrm{Vol}_n(K + t B) = \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, 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(tK) = t^{n-j} W_j(K)~, \quad t \geq 0~.

Statement[edit]

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

v(S) = \sum_{j=0}^n c_j W_j(S)~.

Corollary[edit]

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

References[edit]

An account and a proof of Hadwiger's theorem may be found in

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. MR 2057247.