Some properties of Gurarii's modulus of convexity (Q1293311)

Let \(E\) be a normed linear space, \(S_E\) and \(B_E\) be the unit sphere and unit ball of \(E\), respectively. In the present paper the authors study properties of \textit{V. I. Gurarii's} modulus of convexity [Mat. Issled. 2, No. 1(3), 141-148 (1967; Zbl 0232.46024)] defined by the formula for \(0\leq\varepsilon\leq 2\), \[ \beta_E(\varepsilon)= \inf\Biggl\{1- \inf_{t\in[0,1]}\|tx+ (1- t)y\|: x,y\in S_E, \|x-y\|=\varepsilon\Biggr\}. \] The authors show that Gurarii's modulus of convexity \(\beta_E(\varepsilon)\) can be defined in several equivalent forms and has analogous properties on the well-known Clarkson's modulus of convexity [see: \textit{K. Goebel} and \textit{W. A. Kirk}, ``Topics in metric fixed point theory'', Cambridge Univ. Press (1990; Zbl 0708.47031)]. They proved that in \(E= \ell^p\), \(p>2\), \(\beta_E(\varepsilon)= 1-(1-(\varepsilon/2)^p)^{1/p}\).