The notion of transversality of the intersection of two smooth manifolds is extended to the Banach space setting, with many applications to variational analysis and optimization. It is closely related to that of regularity for sets – while regularity refers to individual sets, transversality can be viewed as metric regularity of the intersection of sets (see, for instance, the book [A. D. Ioffe, Variational analysis of regular mappings. Theory and applications. Cham: Springer (2017; Zbl 1381.49001)]). There are several equivalent characterizations of transversality, among these, we mention only the metric characterization given by A. D. Ioffe: two closed subsets \(A,B\) of a Banach space \(X\) are transversal at a point \(x_0\in A\cap B\) if there exists \(K,\delta>0\) such that \((*)\; d(x,(A-a)\cap(B-b))\le K(d(x,A-a)+ d(x,B-b))\) for all \(x\in x_0+\delta \bar B_X\) and all \(a,b\in\delta\bar B_X\), where \(\bar B_X\) denotes the closed unit ball of \(X\). They are called substransversal at \(x_0\) if, instead of \((*)\), \(d(x,A\cap B)\le K(d(x,A)+ d(x,B))\) holds.
Other related notions – tangential transversality, strong tangential transversality – were introduced and studied in several previous papers by N. Ribarska (in cooperation with S. Apostolov, M. Bivas, M. Krastanov), where it was shown that both transversality and strong tangential transversality imply tangential transversality, which, in its turn, implies subtransversality, but the relation between transversality and strong tangential transversality has not been clarified. The aim of this paper is to prove that strong tangential transversality implies transversality, that the reverse implication fails and the two notions coincide for convex sets. The obtained results are applied to prove some general sum rules for Clarke subdifferentials as well as to some results on closed convex bounded subsets \(A,B\) of a Banach space \(X\): if \(A-B\) is dense in a neighborhood of 0, then the closure of \(A-B\) contains a neighborhood of 0.