Given a closed, non necessarily convex set D of an Hilbert space, we consider the problem of the existence of a neighborhood U on which the projection on D is uniquely defined and lipschitz continuous, and such that the corresponding minimization problem has no local minima. After having equipped the set D with a family P of pathes playing for D the role the segments play for a convex set, we define the notion of strict quasi convexity of (D, P), which shall ensure the existence of such a neighborhood U. Two constructive sufficient condition for the strict-quasiconvexity of D are given, the RG-size x curvature condition and the Q-size x curvature condition, which both amount to checking for the strict positivity of quantities defined by simple formulas in terms of arc length, tangent vectors and radii of curvatures along all pathes of P. An application to the study of well- posedness and local minima of a non-linear least square problem is given.