Hadwiger's theorem

In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem states that the space of "measures" (see below) defined on finite unions of compact convex sets in R^"n" consists of one "measure" that is "homogeneous of degree "k" for each "k" = 0, 1, 2, ..., "n", and linear combinations of those "measures".

Here "measure" means a real-valued function "m" that is invariant under rigid motions (combinations of rotations and translations), finitely additive (if "A" and "B" are finite unions of compact convex sets then "m"("A" ∪ "B") = "m"("A") + "m"("B") − "m"("A" ∩ "B"), and "m"(∅) = 0), and convex-continuous (its restriction to convex sets is continuous with respect to the Hausdorff metric). The countable additivity condition that is usually a part of the definition of measure is not required here.

"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 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.