Some moduli and constants related to metric fixed point theory (Q2761630)

The most common way for creating a lot of quantitative descriptions of geometrical properties of Banach spaces is to define a real function (a modulus) and from this define a suitable constant or coefficient closely related to this function. The aim of this appendix to the article by \textit{St. Prus} [ibid. 93-132 (2001; Zbl 1018.46010)] is to give only a brief summary of a few of those properties that are in some ways related to metric fixed point theory.NEWLINENEWLINENEWLINE\textit{J. A. Clarkson} [Trans. Am. Math. Soc. 40, 396-414 (1936; Zbl 0015.35604)] introduced the modulus of convexity \(\delta_x\) and related coefficient \(\varepsilon_0(x)\) to define uniformly convex spaces. A Banach space \((X,\|\cdot \|)\) is said to be uniformly convex whenever \(\delta_x (\varepsilon) >0\) for a \(a<\varepsilon\leq 2\), or equivalently, if \(\varepsilon_0 (x)=0\), where \(\delta_x: [0,2]\to [0,1]\) is given by \(\delta_x (\varepsilon)= \inf\{1- \tfrac 12 \|x+y\|: x,y\in B_x\), \(\|x-y\|\geq \varepsilon\} \), \(\varepsilon_0 (x)=\sup\{ \varepsilon\in [0,2]: \delta_x (\varepsilon)= 0\}\), where \(B_x\) is the unit ball in the space \(x\). A great number of moduli have been defined since then.NEWLINENEWLINENEWLINEIn Section 2 of this article, the author lists some facts and geometrical properties in terms of modulus and/or its coefficient. The moduli are the Clarkson modulus of convexity, the Lovaglia local modulus of convexity, Smulian's modulus of weak uniform rotundity, the Smulian modulus of weak* uniform rotundity, Lovaglia's modulus of weak local uniform rotundity, the Lindenstrauss modulus of smoothness, Gurarii's modulus of convexity, the Geremia-Sullivan modulus of \(k\)-rotundity, Milman's \(k\)-dimensional modulus of convexity and smoothness, Opial's modulus, and a few others.NEWLINENEWLINENEWLINEIn Section 3, the author lists properties in terms of the Jung constant, Bynum's coefficient of normal structure, the Gao-Lau coefficient, the coefficient for the semi-Opial property, and in terms of a few others. A list of 93 references is given at the end. The article is a good survey on the subject.NEWLINENEWLINEFor the entire collection see [Zbl 0970.54001].