Convex trigonometry with applications to sub-Finsler geometry

Abstract

Abstract A new convenient method for describing flat convex compact sets and their polar sets is proposed. It generalizes the classical trigonometric functions and . It is apparent that this method can be very useful for an explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Finsler problems with two-dimensional control lying in an arbitrary convex set is investigated. Namely, problems on the Heisenberg, Engel, and Cartan groups and also Grushin’s and Martinet’s cases are considered. Particular attention is paid to the case when is a convex polygon. Bibliography: 13 titles.