Convex trigonometry

I will talk about a new convenient method for describing plane convex compact sets and their polars, which generalizes the classical trigonometric functions $\cos$ and $\sin$. Properties of this pair of functions in the case of the unit circle are inherited by two pairs of functions $\cos_\Omega$, $\sin_\Omega$ and $\cos_{\Omega^\circ}, \sin_{\Omega^\circ} $ constructed for the set $\Omega$ and its polar $\Omega^\circ$. This method has proven to be very useful for explicitly describing solutions of optimal control problems with two-dimensional control. With its help, in 2018, I was able to explicitly find geodesics in a series of subfinsler problems for the cases of Heisenberg, Grushin, Martin, Engel, and Cartan. In 2019, together with Yu. L. Sachkov and A. A. Ardentov we explicitly solved more than 10 classical problems. For example, in the talk I will show Finsler geodesics on the Lobachevsky plane.