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

Ramasinghe, W.

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 \(\ell^{k+1}_1\).

Reviewer: Barry Turett (Rochester)

MSC:

46B20

Geometry and structure of normed linear spaces

Keywords:

Banach space; common modulus of convexity; \(k\)-dimensional area; \(k\)-uniformly convex; \(k\)-uniformly rotund; local \(n\)-structure; nonreflexive Banach space; normal structure; modulus of convexity; modulus of rotundity; modulus of \(k\)-rotundity; reflexive Banach space; superreflexive Banach space; uniformly nonoctahedral

Citations:

Zbl 0063.01058; Zbl 0472.46018

References:

[1]	Davis, W.; Johnson, W.; Lindenstrauss, J., Studia Math., 55, 2, 122-139 (1976) · Zbl 0344.46031
[2]	Day, M. M., Ann. of Math., 45, 2, 375-385 (1944) · Zbl 0063.01058
[3]	Geremia, R.; Sullivan, F., Ann. Mat. Pura Appl., 127, 231-251 (1981) · Zbl 0472.46018
[4]	James, R. C., Israel J. Math., 18, 145-155 (1974) · Zbl 0292.46014
[5]	Milman, V. D., Uspekhi Mat. Nauk, 26, 6, 73-149 (1971)
[6]	Lin, P-K, J. Math. Anal. Appl., 132, 2, 349-355 (1988) · Zbl 0649.46014
[7]	Ramasinghe, W., Bull. Sci. Math., 147, 1-6 (2018) · Zbl 1407.46012
[8]	Silverman, E., Rev. Mat. Univ. Parama, 2, 47-96 (1951)
[9]	Sullivan, F., Canad. J. Math., 31, 3, 628-636 (1979) · Zbl 0422.46011
[10]	Yu Xin Tai, Zang Er bin, Liu Zheng, “On K-UR Banach Spaces”, J. East China Normal University, 1981, issue 1, 1-8.
[11]	Suyalatu Wulede, Tingting Li, Xuan Qin, J. Funct. Spaces, 2015, 865305.
[12]	Cepedello Boiso, M., J. Funct. Anal., 191, 1, 1-16 (2002) · Zbl 1010.46010
[13]	Veena Sangeetha, M., Bull. Aust. Math. Soc., 99, 2, 262-273 (2019) · Zbl 1417.46013
[14]	W. Ramasinghege, Multidimensional geometric moduli and exterior algebra of a Banach space, PhD Thesis, The Ohio State University, 1988.