Smoothness problem of sub-Riemannian geodesics

The topic of the talk lies at the intersection of smooth manifold geometry and interpolation theory. The most intriguing object of sub-Riemannian geometry is abnormal geodesics. These geodesics determine the local structure of the metric, but do not satisfy an analogue of the equation with Christoffel's symbols. Two main problems of sub-Riemannian geometry are related to them - the Sard conjecture and the problem of their smoothness. Interestingly, the fact of the existence of abnormal geodesics has long been questioned by the community until Montgomery provided an explicit counterexample on the Martinet distribution in 1991.
At the seminar, I will speak about a new approach to the problem: together with M. I. Zelikin, we managed to show that the properties of smoothness and minimum length for a curve are dual from the point of view of convex geometry. This approach allows us to prove $L_p$-Hölder continuity of velocity on geodesics.
No preliminary knowledge of sub-Riemannian geometry is required.