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Hadwiger's theorem
From Wikipedia, the free encyclopedia
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 Rn 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 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 ck. 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.
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.