Multi-polytopes and the BKK theorem for quasitoric bundles

Quasitoric bundles are one of the easiest objects in parametric toric topology. These are locally trivial fiber bundles with a quasitoric manifold as fiber. In my talk I will discuss the connection between quasitoric bundles and geometry of polytopes. First I will talk about the theory of multi-polytopes introduced by Masuda and their connection to quasitoric manifolds. Then I will discuss a class of smooth polynomial measures on multi-polytopes. I will finish the talk with a generalisation of the classical BKK theorem to the case of quasitoric bundles. This result connects the intersection numbers in cohomology rings of quasitoric bundles to the geometry of multi-polytopes. The talk is based on joint works with Askold Khovanskii and Ivan Limonchenko.