Editing Hadwiger's theorem (section)

== 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>