Typical Convexity (Concavity) of Dini-Hadamard Upper (Lower) Directional Derivatives of Functions on Separable Banach Spaces

By "typical" we mean "valid outside a small (or negligible) set". There are various concepts of "smallness" used in analysis: measure theoretic (null sets of different kind), topological (sets of the first Baire category), metric (σ-porous sets or directionally σ-porous sets), analytic (countable unions of sets that can be represented as (subsets of) graphs of certain classes of Lipschitz functions). Here we basically deal with two types of small sets associated with the two last types of smallness concepts, namely directionally σ-porous sets and so called sparse sets. These two classes of sets are among the smallest: directionally σ-porous sets are sets of the first Baire category and at the same time Aronszajn null (hence Haar null, hence sets of Lebesgue measure zero if the space is finite dimensional). In turn, sparse sets is a proper subclass of the class of directionally σ-porous sets