Summary. Two kinds of s-convexity (0 < s < 1) are discussed. It is proved among others that s-convexity in the second sense is essentially stronger than the s-convexity in the first, original, sense whenever 0 < s < 1. Some properties of s-convex functions in both senses are considered and various examples and counterexamples are given.
Abstract. Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant C NJ (X), and the normal structure coefficient N (X) of Banach spaces X are investigated. Relations between J(X) and J(X * ) are given as an answer to a problem of Gao and Lau [16]. Connections between C NJ (X) and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the C NJ (X)-constant, which implies that a Banach space with C NJ (X)-constant less than 5/4 has the fixed point property. . In particular, they determined or estimated C NJ (X) for various spaces X, and showed that some properties of X such as uniform non-squareness, superreflexivity or type and cotype can be described in terms of the constant C NJ (X).The aim of this paper is to clarify some relations beween the C NJ (X)-constant and some other geometrical constants, especially the non-square constant J(X) of James and the normal structure coefficient N (X). In addition, we investigate the James constant J(X) more carefully. Everything is supported by several examples of concrete Banach spaces with the calculation of these constants.The paper is organized as follows: In Section 1 we collect necessary properties of the modulus of convexity and modulus of smoothness. In Section 2 the uniformly non-square spaces and the James constant J(X) are consid-2000 Mathematics Subject Classification: 46B20, 46E30, 46A45, 46B25.
The structure of the Cesàro function spaces Ces p on both [0, 1] and [0, ∞) for 1 < p ∞ is investigated. We find their dual spaces, which equivalent norms have different description on [0, 1] and [0, ∞). The spaces Ces p for 1 < p < ∞ are not reflexive but strictly convex. They are not isomorphic to any L q space with 1 q ∞. They have "near zero" complemented subspaces isomorphic to l p and "in the middle" contain an asymptotically isometric copy of l 1 and also a copy of L 1 [0, 1]. They do not have Dunford-Pettis property but they do have the weak Banach-Saks property. Cesàro function spaces on [0, 1] and [0, ∞) are isomorphic for 1 < p ∞. Moreover, we give characterizations in terms of p and q when Ces p [0, 1] contains an isomorphic copy of l q .
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