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

Abstract

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 on $R^2$, 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 ${\int }_0^{1}I\left(h^{\prime }\left(t\right)\right)dt$ 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).