In 1952, Hugo Hadwiger established a remarkable classification result.
He proved that all rigid motion-invariant, continuous valuations on the space
of convex bodies (compact convex sets) in Rn are linear combinations of
intrinsic volumes. Here, a functional Z defined on convex bodies is called a
valuation (or additive) if
Z(K) + Z(L) = Z(K ∪ L) + Z(K ∩ L)
for all convex bodies K and L such that K ∪ L is again a convex body.
In R3, the Hadwiger theorem states that for every rigid motion-invariant,
continuous valuation Z, there are c0, c1, c2, c3 ∈ R such that
Z(K) = c0V0(K) + c1V1(K) + c2V2(K) + c3V3(K)
for every convex body in R3. Here, V0(K) = 1 (the Euler characteristic)
and V3(K) is the 3-dimensional volume of K, while V2(K) is (up to a constant
multiple) the perimeter of K and V1(K) its mean width.
We will discuss this result, some of its consequences and applications,
and some of the many results it inspired. In particular, we will describe a
recent functional version of the Hadwiger theorem (joint work with Andrea
Colesanti and Fabian Mussnig).