The Ptolemy and Zbăganu constants of normed spaces
It is known that inner product spaces are characterized by Ptolemy's inequality: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. That is, for any points \(x,y,z,w\) in the space, \(\| x-y\| \| z-w\| \leq \| x-z\| \| y-w\| + \| z-y\| \| x-w\|\). It is also known [cf.\ \textit{Y.\,Pinchover, S.\,Reich} and \textit{I.\,Shafrir}, ``The Ptolemy Constant of a Normed Space'', Am.\ Math.\ Mon. 108, 475--476 (2001; Zbl 1196.46016)] that, for any normed linear space \(X\), there exists a constant \(1\leq C\leq 2\) such that, for any \(x\), \(y\), \(z\), \(w\) in \(X\), \(\| x-y\| \| z-w\| \leq C(\| x-z\| \| y-w\| + \| z-y\| \| x-w\|)\). The smallest such \(C\) is the Ptolemy constant \(C_P(X)\) of \(X\). In the article under review, the authors study the relationship between the Ptolemy constant of a real normed linear space \(X\) and the geometry of the space. In addition to computing the Ptolemy constant of several normed linear spaces, the authors present inequalities relating \(C_P(X)\) to other parameters of a normed linear space, such as the James constant, the characteristic of convexity, the von Neumann-Jordan constant, and the Zbăganu constant \(C_Z(X):= \sup\{ \frac{\| x+y\| \| x-y\|} {\| x\|^{2} + \| y\|^{2}} : x\neq 0, y\neq 0\}\). For example, the authors show that \(C_P(X) \geq C_Z(X)\) and that, if \(C_Z(X) < (1 + \sqrt{3})/2\), then \(X\) has uniform normal structure: therefore, if either \(C_P(X)\) or \(C_Z(X)\) is less than \((1 + \sqrt{3})/2\), \(X\) has the fixed point property for nonexpansive mappings.
scientific article; zbMATH DE number 5696627
scientific article