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Multidimensional moduli of convexity and rotundity in Banach spaces. (English. Russian original) Zbl 1455.46019

Funct. Anal. Appl. 54, No. 1, 59-63 (2020); translation from Funkts. Anal. Prilozh. 54, No. 1, 75-80 (2020).
In [Ann. Math. (2) 45, 375–385 (1944; Zbl 0063.01058)], M. M. Day proved that the \(\ell^p\)-product \((\sum \oplus X_n)_p\), \(1<p<\infty\), of a sequence of Banach spaces is uniformly rotund (\(1\)-UR) if and only if each \(X_n\) is uniformly rotund with a common modulus of convexity. R. Geremia and F. Sullivan [Ann. Mat. Pura Appl., IV. Ser. 127, 231–251 (1981; Zbl 0472.46018)] proved that the same product is \(2\)-UR if and only if all but one of the \(X_n\)’s are \(1\)-UR with a common modulus of convexity and the remaining space is \(2\)-UR.
In the article under review, the author extends the Geremia-Sullivan result to \(k\)-uniform rotund (\(k\)-UR) spaces for integers \(k\geq 2\). In particular, the author proves that the product \((\sum\oplus X_n)_p\) is \(k\)-UR (but not \((k-1)\)-UR) if and only if there exist integers \(j\geq 1\) and \(n_1,\dots, n_j\) such that, if \(k=n_1+\cdots+n_j - (j-1)\) and \(2\leq n_i\leq k\) for \(i=1,\dots,j\), the \(X_n\)’s can be reindexed so that \(X_i\) is \(n_i\)-UR (but not \((n_i-1)\)-UR) for \(i=1,\dots,j\), and the remaining \(X_n\)’s are \(1\)-UR with a common modulus of convexity. The author also proves that a \(k\)-UR Banach space cannot contain arbitrarily close copies of the positive face of the unit ball of \(\ell^{k+1}_1\).

MSC:

46B20 Geometry and structure of normed linear spaces
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References:

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