Semidefinite programming for optimizing convex bodies under width constraints

Abstract

We consider the problem of minimizing a functional (such as the area, perimeter and surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér-Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of constant-width planar bodies, rotors and space bodies of revolution are revisited. The approach seems promising to investigate more difficult optimization problems in the class of three-dimensional convex bodies. © 2012 Copyright Taylor and Francis Group, LLC.