Almost flat locally finite coverings of the sphere

For any subset A of the unit sphere of a Banach space X and for ε∈[0,2) the notion of ε-flatness is introduced as a “measure of non-flatness” of A. For any positive ε, construction of locally finite tilings of the unit sphere by ε-flat sets is carried out under suitable ε-renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.