The isoperimetric problem for convex hulls and large deviations rate functionals of random walks

We study the asymptotic behaviour of the most likely trajectories of a planar random walk that result in large deviations of the area of their convex hull. If the Laplace transform of the increments is finite, such a scaled limit trajectory h solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length of h is replaced by the large deviations rate functional ∫ 0 1 I ( h ′ ( t ) ) d t and I is the rate function of the increments. Assuming that the distribution of increments is not supported on a half-plane, we show that the optimal trajectories are convex and satisfy the Euler–Lagrange equation, which we solve explicitly for every I . The shape of these trajectories resembles the optimizers in the isoperimetric inequality for the Minkowski plane, found by Busemann (1947).