Hadwiger's theorem

In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in Rⁿ consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and linear combinations of those "measures".

"Homogeneous of degree k" means that rescaling any set by any factor c > 0 multiplies the set's "measure" by c^k. The one that is homogeneous of degree n is the ordinary n-dimensional volume. The one that is homogeneous of degree n − 1 is the "surface volume." The one that is homogeneous of degree 1 is a mysterious function called the mean width, a misnomer. The one that is homogeneous of degree 0 is the Euler characteristic.

The theorem was proved by Hugo Hadwiger, and led to further work on intrinsic volumes.

References

An account and a proof of Hadwiger's theorem may be found in Introduction to Geometric Probability by Daniel Klain and Gian-Carlo Rota, Cambridge University Press, 1997.