Vysotskiy - The Isoperimetric Problem for Convex Hulls and the Large Deviations Rate Functions

Explore a 28-minute lecture on the large deviations principle for the area of convex hulls in planar random walks. Delve into the asymptotic shape of the most likely trajectories resulting in such large deviations. Learn about the anisotropic inhomogeneous isoperimetric problem for convex hulls, where traditional length is replaced by the large deviations rate functional. Discover how optimal trajectories are smooth, convex, and satisfy the Euler-Lagrange equation when the distribution of increments is not contained in a half-plane. Examine the explicit solution for every rate function I, drawing parallels to Busemann's 1947 solution for the isoperimetric problem in the Minkowski plane. Gain insights into advanced mathematical concepts presented by the Hausdorff Center for Mathematics, bridging probability theory, geometry, and optimization.