On subdifferentials of a minimal time function in Hausdorff topological vector spaces

In a general Hausdorff topological vector space E, we study the minimal time function associated with a nonempty closed (both cases: convex and nonconvex) set S and a bounded closed convex set Omega defined by T-S,T-Omega (x) := inf {t >= 0 : S boolean AND (x + t Omega) not equal empty set}. We prove and extend various important properties on directional derivatives and subdifferentials of T-S,T-Omega at points in S. These results are used to prove various new characterizations of the convex tangent cone, the convex normal cone, the Clarke tangent cone, the Bouligand tangent cone, and Clarke normal cone to S in terms of the minimal time function for points inside S, in Hausdorff topological vector spaces. Those characterizations are used to scalarize the tangential regularity of sets, in Hausdorff topological vector spaces, as the directional regularity of T-S,T-Omega. Our results extend various existing results, in convex and nonconvex cases, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces. Even in Banach spaces and in normed spaces our results are new, and they extend various existing results on the distance function to closed sets by taking Omega to be the closed unit ball. Applications to normal and subidfferential regularities in normed vector spaces are also given in the last section of the paper.