Gergely Ambrus: Critical central sections of the cube

Abstract: Volumes of central hyperplane sections of the d-dimensional cube Q_d have been studied for over a century. Following the work of Hensley, and Vaaler, K. Ball proved in 1986 that the sections of maximal volume are normal to the main diagonal of a 2-dimensional face. Sections orthogonal to the main diagonal of a k-dimensional face are called k-diagonal. In 2021, Bartha, Fodor and González Merino proved that the volumes of k-diagonal sections form a monotone increasing sequence for k>=3. One may study the volume of central sections as a function of the normal vector. It has been naturally conjectured that critical directions with respect to the volume function are diagonal. First, we give an analytic characterization of critical directions that has also been proved independently by Ivanov and Tsiutsiurupa. Based on this result, we determine critical sections of Q_d up to d<=4, which show that for d=4, there exist non-diagonal critical directions. Then proceed to prove that, surprisingly, there exist full-dimensional, non-diagonal critical directions for every d>=4. This disproves the above long-standing conjecture. The arguments use Fourier analytic, geometric, stochastic, and combinatorial methods. In particular, one of the key tools is a generalization of asymptotic bounds on second-order Eulerian numbers obtained by Leusier and Nicolas in 1992. Partly joint work with Barnabás Gárgyán.