Bishop–Phelps cones given by an equation in Banach spaces

Abstract

In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a ‘large family’ in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation.