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

<p dir="ltr">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 onR2, such a scaled limit trajectoryh solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length ofh is replaced by the large deviations rate functional∫01I(h′(t))dt andI 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 everyI. The shape of these trajectories resembles the optimizers in the isoperimetric inequality for the Minkowski plane, found by Busemann (1947).</p>