Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Description

The combinatorial type of a polytope is the poset of faces of the polytope ordered by inclusion. We study to what extent the combinatorial type of a polytope determines the combinatorial types of its slices, and, conversely, how much the combinatorial types of slices determine the combinatorics of a polytope. We show that in general, combinatorial types of slices is not even enough to determine the f-vector of a polytope. However, for sufficiently generic simple polytopes, already the function recording the number of vertices on every central slice uniquely determines the combinatorial type of the polytope. We formulate combinatorial analogues of the Busemann-Petty problem and show that they fail in every dimension. This is joint work with Anna Birkemeyer, Marie-Charlotte Brandenburg and Niklas Prün, and started in a REU last summer.