Renormings and extremal structures

Abstract

In this paper we use renorming techniques to settle several questions on extremal structures of Banach spaces. We construct a unitary vector in a dual space which is not a weak^∧ -unitary. We construct an exposed point in the unit ball of a Banach space X that remains exposed in the unit ball of X⁽⁴⁾ but is not extreme in the unit ball of X⁽⁶⁾.