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Journal of Convex Analysis 17 (2010), No. 1, 001--011
Copyright Heldermann Verlag 2010

On Local Milman's Moduli

Stanislaw Prus
Institute of Mathematics, M. Curie-Sklodowska University, 20-031 Lublin, Poland

Mariusz Szczepanik
Institute of Mathematics, M. Curie-Sklodowska University, 20-031 Lublin, Poland

In his paper "Geometric theory of Banach spaces. Part II: Geometry of the unit sphere" [Russian Math. Surveys 26 (1971) 79--163, translation from Usp. Mat. Nauk 26 (1971) 73--149], V. D. Milman gave a scheme of defining moduli which can be used as tools for studying the geometry of Banach spaces. For instance they were used to characterize uniform convexity, uniform smoothness and multi-dimensional counterparts of these properties. Infinite-dimensional Milman's moduli turned out to be related to the Kadec-Klee property, nearly uniform convexity and nearly uniform smoothness. They were successfully applied to some problems of nonlinear analysis, including differentiation of mappings on Banach spaces and metric fixed point problems.
In this paper we study local versions of some Milman's moduli. In particular we show that a two-dimensional Milman's modulus is in a sense equivalent to the standard modulus of smoothness. This is a quantitative version of a result obtained by J. Banas ["On moduli of smoothness of Banach spaces", Bull. Pol. Acad. Sci. 34 (1986) 287--293]. We also establish an inequality between local Milman's moduli corresponding to finite-dimensional and infinite-dimensional smoothness. As a tool we use a formula obtained by W. Kaczor and S. Prus ["Asymptotical smoothness and its applications", Bull. Austral. Math. Soc. 66 (2002) 405--418] which describes the latter modulus in terms of weakly null nets. We also give an analogous formula for the modulus corresponding to infinite-dimensional convexity. An inequality between an infinite-dimensional Milman's modulus of convexity and the standard modulus of convexity was obtained by W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman ["Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math. Soc. (3), 84 (2002) 711--746]. We give an example showing that its local version is not true.

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