Given a closed subset S of a Banach space X, we study the minimal time function to reach a point x of X starting from S. We relate this problem to the study of the geodesic distance from x to S associated with a Riemannian metric or a Finsler metric. It appears that both cases can be treated simultaneously and yield solutions to a Hamilton-Jacobi equation. We detect simple links between the general framework we propose and multivalued semigroups. New concepts of infinitesimal generators of such semigroups are considered.