Research On Problems Related To Generalized Nonsquare Constants

Nonsquare constants represent nonsquareness. They are not only related to uniform formal structure, but also related to some geometrical properties of the underlying spaces. The geometry calculation of geometry constants can give a more precise description of the properties in the spaces. The theories of nonsquare constants are mainly studied in this paper, and a new nonsquare constant is defined as a generalized nonsquare constant.First of all, the development of geometric theory in Banach Spaces and the main research results of nonsquare constants and pointwise nonsquare constants by prede-cessors are introduced. Moreover, the backgroud and significance of this paper is pr-esented.Second, some basic definitions and theories of nonsquare constants and the isosceles orthogorality are introduced. By constructing a Banach space with P_λproperty and calculating the value of nonsquare constants. This indicates that the nonsquare constant is not necessary to be 2^1/2 in some spaces with P_λproperty for someλ, and that P_λproperty does not imply strict convexity or even uniform convexity. Furthermore, uniform nonsquareness of spaces with P-property is shown. The last part of this part devotes to the generalized nonsquare constant introduced .Finally, the uniqueness of thinnability is studied. A new geometrical constant as some kind of generalized nonsquare constant are introduced. It is shown that some kind of generalized nonsquare constant is a increasing function with respect to r , and that the normal space is strictly convex if and only if some kind of generalized nonsquare constant is 2 that is not attainable for every r from -2 to 2.