Volumes of central hyperplane sections of the d-dimensional cube Q_d have been studied for over a century: it is known that minimal sections are parallel to a facet, while K. Ball proved in 1986 that the sections of maximal volume are normal to the main diagonal of a 2-dimensional face. Many of the related results were achieved by using analytic methods: the volumes in question can be expressed by the so-called Laplace-Pólya integrals. These are also in connection with Eulerian numbers of the first kind, and hence provide a connection between geometric estimates and combinatorial inequalities. In our joint work with Barnabás Gárgyán, we establish new bounds for the Laplace-Pólya integrals by entirely combinatorial means, which then imply new results in both fields. In particular, we prove the existence of full-dimensional, non-diagonal critical central sections of the cube in every dimension at least 4, and generalize asymptotic bounds on Eulerian numbers obtained by Leusier and Nicolas in 1992.
Critical Central Sections of the Cube
Gergely Ambrus (U Szeged & Rényi Institute, Hungary)
Tuesday, March 12th, 2pm ET, on Zoom