Convex Trigonometry and Sub-Finsler Geometry

In this presentation, I will discuss a new convenient method for describing the boundaries of flat convex compact sets and their polars, generalizing classical trigonometric functions like $\cos$ and $\sin$. The properties of this pair of functions for the unit circle are inherited by two pairs of functions — for the set itself and its polar. These functions have proven to be very useful for solving so-called sub-Finsler problems. An example of such a problem is the Dido's problem: finding a curve in the plane of minimal length enclosing a given area. When the length of the curve is measured in Euclidean metric, the answer is well-known. However, if the length of the curve is measured, for example, in the $L_p$ metric on the plane $(|x|^p+|y|^p)^{1/p}$ (or any other non-Euclidean metric), the problem immediately becomes much more interesting. In the presentation, I will also provide other illustrative examples.
No prior knowledge of sub-Finsler geometry is required.