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$^\ast$-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 $X^{(6)}$.