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. Jiménez-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.