Starshapedness and Convexity in Carnot Groups and Geometry of Hormander Vector Fields

Abstract

investigate the notion of starshaped sets, also known as starlike sets: we can say that starshaped sets satisfy the same geometric characterisation of convex sets, like level sets and starshaped hull but not w.r.t. all its interior points. Starshaped sets are not yet completely understood in Carnot groups and sub-Riemannian manifolds. So we first consider the nature of starshapedness in Carnot groups. We consider two different notions of starshaped sets in Carnot groups: the first one is called strongly G−starshapedness and the second one is called weakly G−starshapedness by considering, respectively, the anisotropic dilations associated to Carnot groups for the first and the concept of curves with constant horizontal velocity w.r.t. given vector fields for the second one; the second definition thus working also in general sub-Riemannian geometries