Event Details

Valuations in Banach lattices

A real-valued valuation on a class of sets S is a mapping V:S→R with the property V(A∪B)+V(A∩B)=V(A)+V(B), whenever A,B,A∪B,A∩B∈S. This notion is relevant in several aspects of convex geometry; it played a key role in the solution of Hilbert's third problem, asking whether an elementary definition for volume of polytopes was possible. Valuations also provide an abstract framework for dealing with some geometric inequalities and have fruitful applications in integral geometry and geometric probability. When dealing with star sets in Rn, i.e. those whose intersection with each directional ray is a (possibly degenerate) interval containing the origin, the radial functions lead to an analogue of valuations which fits in the language of vector lattices: Namely, a valuation on a vector lattice X is a mapping V:X→R such that for f,g∈X one has V(f∨g)+V(f∧g)=V(f)+V(g). Our aim in this talk is to provide recent results on representations of real-valued valuations on Banach lattices, with particular emphasis on spaces of the form C(K) and Lp(μ).