Arnold's seminar, October 27, 1998.
"The Hodge-Riemann relations for simple convex polytopes"
The set of all polytopes that are analogous to a certain simple polytope can be canonically embedded into a vector space. The function of volume extends to a polynomial on this vector space. There exists a close connection between some properties of the volume polynomial and the combinatorial structure of the polytope. Several results that were first obtained using the geometry of toric varieties, were translated to the geometrical language and proved within the geometry of convex polytopes. For example, the hard Lefschetz theorem and the Hodge-Riemann relations in the cohomology ring of a toric variety carry over to the geometry of polytopes. The main object is a new proof of the Stanley theorem (necessity of McMullen's conditions on the $f$-vector). This proof uses the same geometrical ideas that are in McMullen's proof, but it deals with another (perhaps more covenient) description of the polytope algebra (proposed by Pukhlikov and Khovanskii).