Some Geometrical Properties And Moduli Of Banach Spaces

Numbers of important results on the geometrical properties of generalized Lebesgue spaces, which mainly focus on the properties of the entire spaces and neglect the local properties and the values of geometric constants and moduli, have been obtained in the last seventy years. On the other hand, generalized nonsquare constants were introduced in 1971, on which there has been few result since then.In this paper, some geometrical properties and constants in generalized Lebesgue spaces are studied; some results concerning nonsquare constants are generalized based on the careful research on generalized nonsquare constants.First, the developments of generalized Lebesgue spaces and generalized nonsquare constants are introduced, the main results on these two fields by several researchers are summarized, and, furthermore, preliminaries, background and significance of the content of this paper are presented.Second, in generalized Lebesgue spaces, extreme points and strict convexity are characterized, and an equivalent condition for uniformλproperty is presented. As an immediate corollary, it is shown that strict convexity and uniformλproperty are equivalent in these spaces. Moreover, Clarkson's inequality is generalized, from which it follows that values of nonsquare constants are obtained in generalized Lebesgue spaces where either p~-≥2 or 1＜p~-≤p~+≤2.Finally, equivalent representations, upper and lower bounds of generalized non-square constants and their values in l_p are obtained. The relationships between generalized nonsquare constants and uniform convexity as well as uniform nonsquareness, from which some sufficient conditions for fixed point property follow, are shown. On the other hand, the relationship between J(t, X) and J(t, X~*), the upper and lower bounds of J(t, X)S(t, X), as well as the relationship between J(t, X) and Banach-Mazur distance are presented. The rest part of the paper devotes to the relationship between J(t, X) and uniformly normal structure.