The Massera-Schäffer inequality related to Birkhoff orthogonality in Banach spaces

Abstract

In this paper, we shall consider  a new constant $MS_B(X)$ which is  used to study the Massera-Schäffer inequality related to Birkhoff orthogonality. We use this constant to characterize the  Hilbert space and  also discuss its relations with some geometric properties of Banach spaces, including uniform non-squareness, uniform convexity and uniform smoothness. Furthermore, we provide a study of $MS_B(X)$  in Radon planes. The equivalent form of this constant in Radon planes is established and used to calculate the value of  $MS_B(l_p-l_q)$ ($1<p, q<\infty$,  $\frac{1}{p}+\frac{1}{q}=1$)