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 \R^n. It was proved by
Hugo Hadwiger Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Ge ...
.


Introduction


Valuations

Let \mathbb^n be the collection of all compact convex sets in \R^n. A valuation is a function v : \mathbb^n \to \R such that v(\varnothing) = 0 and for every S, T \in \mathbb^n that satisfy S \cup T \in \mathbb^n, 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(\varphi(S)) = v(S) whenever S \in \mathbb^n and \varphi is either a
translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
or a
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
of \R^n.


Quermassintegrals

The quermassintegrals W_j : \mathbb^n \to \R are defined via Steiner's formula \mathrm_n(K + t B) = \sum_^n \binom 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_ is proportional to the
mean width In geometry, the mean width is a measure of the "size" of a body; see Hadwiger's theorem for more about the available measures of bodies. In n dimensions, one has to consider (n-1)-dimensional hyperplanes perpendicular to a given direction \hat in ...
, and W_n is the constant \operatorname_n(B). W_j is a valuation which is
homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
of degree n - j, that is, W_j(tK) = t^ W_j(K)~, \quad t \geq 0~.


Statement

Any continuous valuation v on \mathbb^n that is invariant under rigid motions can be represented as v(S) = \sum_^n c_j W_j(S)~.


Corollary

Any continuous valuation v on \mathbb^n that is invariant under rigid motions and homogeneous of degree j is a multiple of W_.


See also

* *


References

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 * {{cite journal, title=A simplified elementary proof of Hadwiger's volume theorem, journal=Geom. Dedicata, volume=105, year=2004, pages=107–120, last=Chen, first=B., mr=2057247, doi=10.1023/b:geom.0000024665.02286.46 Integral geometry Theorems in convex geometry Theorems in probability theory