Schäffer-type constant and uniform normal structure in Banach spaces (Q306963)

The Schäffer constant is defined by \[ S(X)=\inf\{\max\{\| x+y\| , \| x-y\| \}: x,y\in S_X\}\}. \] In [Proceedings of the 2nd international symposium on Banach and function spaces II (ISBFS 2006), Kitakyushu, Japan, September 14--17, 2006. Yokohama: Yokohama Publishers. 191--220 (2008; Zbl 1163.46012)], \textit{Y. Takahashi} introduced the Schäffer-type constant \(S_{X,t}(\tau)\) as a generalization of the previous coefficient \(S(X)\). For \(\tau\geq 0\) and \(1<t\leq \infty\), the constant \(S_{X,t}(\tau)\) is defined by \[ S_{X,t}(\tau)=\inf\{{\mathcal M}_t(\| x+\tau y\| , \| x-\tau y\| ): x,y\in S_X\}, \] where \(\mathcal{M}_t(a,b)\) is the generalized mean defined by \[ \mathcal{M}_t(a,b):=\left({a^t+b^t\over 2}\right)^{1/t}, \quad a,b\geq 0. \] The value \(S_{X,\tau}(1)\) is related to the uniform nonsquareness and it can be computed by using the modulus of smoothness \(\rho_X(\epsilon)\) [\textit{F.-H. Wang} and \textit{C.-S. Yang}, Stud. Math. 201, No. 2, 191--201 (2010; Zbl 1208.46019)]. In the present paper, the authors give a method to determine and estimate the value of \(S_{X,\tau}(1)\) for absolute normalized norms on \(\mathbb{R}^2\) that it is independent of the knowledge of \(\rho_X(\epsilon)\). Some sufficient conditions implying uniform normal structure are also presented.