We analyse the structure of finite dimensional subspaces of the set of points of strong subdifferentiability in a dual space. In a dual L1(μ) space, such a subspace is in the discrete part of the Yoshida-Hewitt type decomposition. In this set up, any Banach space consisting of points of strong subdifferentiability is necessarily finite dimensional. Our results also lead to streamlined and new proofs of results from the study of strong proximinality for subspaces of finite co-dimension in a Banach space. © 2021 Heldermann Verlag.