Tangency, paratangency and four-cones coincidence theorem in Carnot groups (Q2922482)

The aim of this paper is to extend some results from nonsmooth analysis concerning tangent cones to Carnot groups. As the authors say: \, ``a Carnot group \(\mathbb G\), endowed with a left-invariant metric \(d_{cc}\) (called Carnot-Carathéodory metric), provides an example of a metric space that is not equivalent to the Euclidean, even locally. At the same time, it still has a sufficiently rich underlying structure (existence of a family of left translations and dilations compatible with \(d_{cc}\)) to give rise to many notions of classical analysis'' (e.g. differentiability and Withney's extension theorem).NEWLINENEWLINEContinuing this line of investigation, the authors consider the analogues of upper and lower tangent cones (Tan\(^+_{\mathbb G}(S,P)\) and Tan\(^-_{\mathbb G}(S,P)\)) and the corresponding paratangent cones (pTan\(^+_{\mathbb G}(S,P)\) and pTan\(^-_{\mathbb G}(S,P)\)) at a point \(P\) of a subset \(S\) of \(\mathbb G\) and study their relationships with differentiability and uniform differentiability. The main result of the paper (Theorem 1.2) gives a characterization of regular intrinsic submanifolds of \(\mathbb G\) as those closed connected subsets \(S\) of \(\mathbb G\) for which these cones coincide at every point in \(S\).NEWLINENEWLINEApplications are given to optimization problems for differentiable functions on Carnot groups---Peano's regula and Lagrange multipliers.