Renormings and extremal structures (Q1766868)

The idea of extending the theory of numerical ranges from Banach algebras to general Banach spaces goes back to \textit{J. Martínez-Moreno, J. F. Mena-Jurado, R. Payá-Albert} and \textit{A. Rodríguez-Palacios} [Ill. J. Math. 29, No. 4, 609--626 (1985; Zbl 0604.46052)]. This theory is applied here to give examples of Banach spaces with interesting extremal structure. In particular, there is a Banach space whose unit ball contains an exposed point, which is also exposed in the fourth dual, but is not even an extreme point in the sixth dual. Another example is a functional in a dual space which does not attain its norm, although its support functionals algebraically span the bidual.