The Degree of Convex Nondensifiability in Banach Spaces

It is shown in this paper that the notion of α-dense curve in a bounded set of a metric space produces a degree of nondensifiability $\phi_d$ which extended to convex hulls of bounded sets in a Banach space generates a measure of noncompactness (briefly MNC) Φ that we will call degree of convex nondensifiability. We also prove that Φ dominates any other MNC and, in particular, dominates the MNC of Hausdorff of the best possible form in infinite dimensional Banach spaces.