Hadwiger's theorem

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.

Valuations
Let $$\mathbb{K}^n$$ be the collection of all compact convex sets in $$\R^n.$$ A valuation is a function $$v : \mathbb{K}^n \to \R$$ such that $$v(\varnothing) = 0$$ and for every $$S, T \in \mathbb{K}^n$$ that satisfy $$S \cup T \in \mathbb{K}^n,$$ $$v(S) + v(T) = v(S \cap T) + v(S \cup T)~.$$

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 $$S \in \mathbb{K}^n$$ and $$\varphi$$ is either a translation or a rotation of $$\R^n.$$

Quermassintegrals
The quermassintegrals $$W_j : \mathbb{K}^n \to \R$$ 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, $$W_0$$ is the volume, $$W_1$$ is proportional to the surface measure, $$W_{n-1}$$ is proportional to the mean width, and $$W_n$$ is the constant $$\operatorname{Vol}_n(B).$$

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

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