Multidimensional Moduli of Convexity and Rotundity in Banach Spaces
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ltidimensional Moduli of Convexity and Rotundity in Banach Spaces W. Ramasinghe Received July 22, 2018; in final form, December 5, 2019; accepted December 17, 2019
Abstract. Geremia and Sullivan [Ann. Math Pure Appl. 127 (1981), 231–251] gave a necessary and sufficient condition for an p -product of spaces to be 2-uniformly rotund. We extend this result to k-uniform rotundity for any integer k > 1. The nonreflexive, uniformly nonoctahedral Banach constructed by James [Israel J. Math. 18 (1974), 145–155] does not contain arbitrarily space X close copies of k+1 , although it is not k-uniformly rotund (k-UR) for any k 2. This shows that 1 a Banach space X not being k-UR does not imply that X contains arbitrarily close copies of k+1 1 for each k 2. We show that a sufficient condition that a Banach space X be not k-UR is that it contains an arbitrarily close copy of one of the faces of k+1 rather than k+1 itself. 1 1 Key words: 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. DOI: 10.1134/S0016266320010086
1. Introduction. The modulus of k-rotundity of a Banach space X is defined based on Silverman’s notion [8] of the k-dimensional area spanned by k + 1 vectors. Sullivan [9] showed that if X is k-uniformly rotund (k-UR) for some k, then X is superreflexive and has normal structure. Consequently, if X is k-uniformly rotund for some k, then X cannot have local n-structure for any n [1]. Geremia and Sullivan [3] proved that 2-uniform rotundity is equivalent to 2-uniform convexity as defined by Milman [5]. Pei-Kee Lin [6] extended this result to k-UR spaces. The author has given another proof of this result [7], [14]. A necessary and sufficient condition for an p -product of spaces to be 2-UR was given in [3]. In the present paper we give a necessary and sufficient condition for an p -product of spaces to be k-UR for every integer k 2. This gives a method for constructing a large class of nontrivial examples of spaces that are k-UR but not (k − 1)-UR. constructed by James [4] is nonreflexive and uniformly nonoctahedral. The Banach space X does not contain arbitrarily close copies of 3 , Therefore, X is not k-UR for any k 2 [4]. Since X 1 does not contain arbitrarily close copies of k+1 for any k 2. This shows that a it follows that X 1 for Banach space X not being k-UR does not imply that X contains arbitrarily close copies of k+1 1 each k 2. In the remaining part of this section, we recall some of the definitions and notation to be used in the sequel. The k-dimensional area A(x1 , . . . , xk , xk+1 ) spanned by vectors x1 , . . . , xk , xk+1 ∈ X is defined as follows: ⎧ ⎫ 1 . . . 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨f1 (x1 ) . . . f1 (xk+1 ) ⎬ : f A(x1 , . . . , xk , xk+1 ) ≡ sup . ∈ B , i = 1, . . . , k . i .. X .. ⎪ .. ⎪ . . ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ fk
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