Solution to the Inverse Wulff Problem by Means of the Enhanced Semidefinite Relaxation Method

Abstract

We propose a novel method of resolving the optimal anisotropy function. The idea is to construct the optimal anisotropy function as a solution to the inverse Wulff problem, i.e. as a minimizer for the anisoperimetric ratio for a given Jordan curve in the plane. It leads to a nonconvex quadratic optimization problem with linear matrix inequalities. In order to solve it we propose the so-called enhanced semidefinite relaxation method which is based on a solution to a convex semidefinite problem obtained by a semidefinite relaxation of the original problem augmented by quadratic-linear constraints. We show that the sequence of finite dimensional approximations of the optimal anisoperimetric ratio converges to the optimal anisoperimetric ratio which is a solution to the inverse Wulff problem. Several computational examples, including those corresponding to boundaries of real snowflakes and discussion on the rate of convergence of numerical method are also presented in this paper.