Editing Hadwiger's theorem

{{Short description|Theorem in integral geometry}}
In [[integral geometry]] (otherwise called geometric probability theory), '''Hadwiger's theorem''' characterises the [[Valuation (geometry)|valuations]] on [[Convex body|convex bodies]] in <math>\R^n.</math> It was proved by [[Hugo Hadwiger]].

== Introduction ==

=== Valuations ===

Let <math>\mathbb{K}^n</math> be the collection of all compact convex sets in <math>\R^n.</math> A '''valuation''' is a function <math>v : \mathbb{K}^n \to \R</math> such that <math>v(\varnothing) = 0</math> and for every <math>S, T \in \mathbb{K}^n</math> that satisfy <math>S \cup T \in \mathbb{K}^n,</math>
<math display=block>v(S) + v(T) = v(S \cap T) + v(S \cup T)~.</math>

A valuation is called continuous if it is continuous with respect to the [[Hausdorff metric]]. A valuation is called invariant under rigid motions if <math>v(\varphi(S)) = v(S)</math> whenever <math>S \in \mathbb{K}^n</math> and <math>\varphi</math> is either a [[Translation (geometry)|translation]] or a [[Rotation (mathematics)|rotation]] of <math>\R^n.</math>

=== Quermassintegrals ===
{{main|quermassintegral}}

The quermassintegrals <math>W_j : \mathbb{K}^n \to \R</math> are defined via Steiner's formula
<math display=block>\mathrm{Vol}_n(K + t B) = \sum_{j=0}^n \binom{n}{j} W_j(K) t^j~,</math>
where <math>B</math> is the Euclidean ball. For example, <math>W_0</math> is the volume, <math>W_1</math> is proportional to the [[Minkowski content|surface measure]], <math>W_{n-1}</math> is proportional to the [[mean width]], and <math>W_n</math> is the constant <math>\operatorname{Vol}_n(B).</math>

<math>W_j</math> is a valuation which is [[Homogeneous function|homogeneous]] of degree <math>n - j,</math> that is,
<math display=block>W_j(tK) = t^{n-j} W_j(K)~, \quad t \geq 0~.</math>

== Statement ==

Any continuous valuation <math>v</math> on <math>\mathbb{K}^n</math> that is invariant under rigid motions can be represented as
<math display=block>v(S) = \sum_{j=0}^n c_j W_j(S)~.</math>

=== Corollary ===

Any continuous valuation <math>v</math> on <math>\mathbb{K}^n</math> that is invariant under rigid motions and homogeneous of degree <math>j</math> is a multiple of <math>W_{n-j}.</math>

== See also ==

* {{annotated link|Minkowski functional}}
* {{annotated link|Set function}}

== References ==

{{reflist}}
{{reflist|group=note}}

An account and a proof of Hadwiger's theorem may be found in 
* {{cite book|mr=1608265|last=Klain|first=D.A.|last2=Rota|author2-link=Gian-Carlo Rota|first2=G.-C.|title=Introduction to geometric probability|url=https://archive.org/details/introductiontoge0000klai|url-access=registration|publisher=Cambridge University Press|location=Cambridge|year=1997|isbn=0-521-59362-X}}

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&ndash;120|last=Chen|first=B.|mr=2057247|doi=10.1023/b:geom.0000024665.02286.46}}

[[Category:Integral geometry]]
[[Category:Theorems in convex geometry]]
[[Category:Theorems in probability theory]]