On Clarkson inequalities and geometry of Banach spaces (Q2741572)

In the paper that introduced the notion of uniform convexity, \textit{J. A. Clarkson} [Trans. Am. Math. Soc. 40, 396-414 (1936; Zbl 0015.35604)] established the following inequalities for the spaces \(\ell_p\). \(L_p\): NEWLINE\[NEWLINE \|x+y\|^p + \|x-y\|^p \leq 2(\|x\|^p + \|y\|^p) \quad \text{for } 1 < p \leq 2,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\|x+y\|^p + \|x-y\|^p \leq 2^{p-1}(\|x\|^p + \|y\|^p) \quad \text{for } 2 \leq p < \infty.\tag{2}NEWLINE\]NEWLINE In this paper the author characterizes those Banach spaces \(X\) that satisfy either inequality (1) or (2) in terms of norm properties of what are here called \(X\)-valued Walsh-Paley martingales. NEWLINENEWLINENEWLINEThere are some minor misprints: for example, reference [9] should probably be [8] (there is no [9]); what is defined as \(S^{(p)}(f)\) is probably the same as \(\|S^{(p)}(f)\|\); and the case \(q=\infty\) should perhaps be excluded from Theorems 3 and 5.