Parametrization of a convex optimization problem by optimal control theory and proof of a Goldberg conjecture

Curves which can be rotated freely in an n-gon (that is, a regular polygon with n sides) so that they always remain in contact with every side of the n-gon are called rotors. The problem of finding the rotor with minimal area is considered and is formulated into an optimal control problem using the support function of a convex body. By the Pontryagin maximum principle and an extension of Noether´s Theorem in optimal control theory, extremal controls are computed. As a consequence, a minimizer is necessarily a regular rotor, which proves a conjecture formulated in 1957 by Goldberg (see).