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We study the asymptotic shape of the most likely trajectories of a planarrandom walk that result in large deviations of the area of the convex hull ofthe first $n$ steps of the walk, as $n$ tends to infinity. If the increments ofthe walk have finite Laplace transform, 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 ratefunctional $\int_0^1 I(h'(t)) dt$ with $I$ being the rate function of theincrements. Assuming that the distribution of increments is not contained in ahalf-plane, we show that the optimal trajectories are smooth, convex, andsatisfy the Euler--Lagrange equation, which we solve explicitly for every $I$.Our solution resembles that of the isoperimetric problem in the Minkowski planefound by Busemann (1947).

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