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In this article, we analyse the structure of finite dimensional subspaces ofthe set of points of strong subdifferentiability in a dual space. In a dual$L_1(\mu)$ space, such a subspace is in the discrete part of the Yoshida-Hewitttype decomposition. In this set up, any Banach space consisting of points ofstrong subdifferentiability is necessarily finite dimensional. Our results alsolead to streamlined and new proofs of results from the study of strongproximinality for subspaces of finite co-dimension in a Banach space.

Structure of sets of strong subdifferentiability in dual $L^1$-spaces