Some geometric measures of spheres in Banach spaces

In this paper, we first give relations between Pythagorean parameters and other well-known parameters: the coefficient of weak orthogonality, James and von Neumann-Jordan constants. Consequently, some known results in [J. Gao, On the generalized Pythagorean parameters and the applications in Banach spaces, Discrete Contin. Dyn. Syst. Ser. B, 8 (3) (2007) 557-567; J. Gao, On some geometric parameters in Banach spaces, J. Math. Anal. Appl. 344 (2007) 114-122; A. Jimenez-Melado, E. Llorens-Fuster, S. Saejung, The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces, Proc. Am. Math. Soc. 134 (2006) 355-364] are deduced and strengthened. Secondly we present several sufficient conditions for a Banach space and its dual to have normal structure. Finally, some open questions posed at the end of Gao (2007) are answered in the negative. (C) 2009 Elsevier Inc. All rights reserved.