Valuations on convex bodies

Speaker: Karoly Boroczky

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.