Valuation on convex bodiesAlgebra & Discrete Mathematics
|Karoly Boroczky, Central European University
|Thu, May 27 2021, 10:00AM
First, the talk reviews Dehn's solution of Hilbert's third problem showing that a cube and a regular tetrahedron of the same volume in 3D are not equidissectable. Then valuations on convex bodies are defined and the fundamental properties due to Hadwiger and McMullen are discussed. Typical examples are volume, mean projections, Euler characteristic, number of lattice points in a lattice polytope.
Alesker's theory based on the representation of the general linear group on the Frechet space of continuous translation invariant valuations is introduced. A recent result on U(n) equivariant tensor valuations will highlight the role of representation theory in this subject.
Finally, time permitting, valuations on lattice polytopes are discussed where a comprehensive theory is still missing.