Abstract: A valuation Z on some family of convex bodies is a function
Z into an abelian group such that whenever the union of the convex
bodies C and K is convex, we have
Z(C\cup K)+Z(C\cap K)=Z(K)+Z(C).
These valuations came up first in Dehn's century old solution of
Hilbert's third problem, but their real renaissance started with
Hadwiger's characterization theorem about 50 years ago. It states that
any continuous real valued valuation on n-dimensional convex bodies is
the linear combination of n+1 basic ones, including the volume,
surface area and the Euler characteristic. The talk surveys some old
results, and some new directions.
You can watch the lecture here: