Existence of diametrically complete sets with empty interior in reflexive and separable Banach spaces

Abstract

In this paper we prove that every infinite-dimensional and separable Banach space (X,‖⋅‖X) admits an equivalent norm ‖⋅‖X,1 such that (X,‖⋅‖X,1) has both the Kadec-Klee and the Opial properties. This result also has a quantitative aspect and when combined with the properties of Schauder bases and the Day norm it constitutes a basic tool in the proof of our main theorem: each infinite-dimensional, reflexive and separable Banach space (X,‖⋅‖X) has an equivalent norm ‖⋅‖0 such that (X,‖⋅‖0) is LUR and contains a diametrically complete set with empty interior.