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We study the local geometry of the sub-Finsler structure induced by a sub-Riemannian metric on a 3-dimensional manifold. We provide a description of the upper part of the cut locus of short geodesics, in some non generic cases. [ABSTRACT FROM AUTHOR]

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE 2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part. [ABSTRACT FROM AUTHOR]

We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K in the first Heisenberg group ℍ 1 induced by a convex body K ⊂ ℝ 2 containing the origin in its interior. Associated to ∥ ⋅ ∥ K there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 . Under the assumption that K has C 2 boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K out of the singular set. In the case of non-vanishing mean curvature, the condition that H K be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K dilated by a factor of 1 H K . Based on this we can define a sphere 핊 K with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K starting from (0 , 0 , 0) until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets. [ABSTRACT FROM AUTHOR]