Gergely Ambrus: Cube sections, Eulerian numbers and the Laplace-Pólya integral

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.