Some Problems Of Pointwise Convex Modulus

Convexity is one of the most fundamental properties for normed spaces. Researching this property contributes to revealing the structure of the normed spaces themselve. At present, the discussions of various convexity properties in spaces have already been perfected. In 1936, J.Clarkson first introduced the concept of uniformly convex Banach spaces ,initiated from geometric structure of the unit sphere of Banach spaces to research the properties of Banach spaces, began to research convex theory of Banach spaces. The same year, J.Clarkson introduced the concept of convex modulus while researching uniformly convex spaces, later the relations between numbers of convex modulus and relevant geometric properties had been drawn. Convex modulus depicted overall degree of convexity of the unit sphere of spaces. It was very different for degree of convexity of each point that it would produce very great influence on the whole feature of the spaces. Therefore, the research of pointwise properties showed the great significance. Many geometric properties of spaces may be pointwise, pointwise geometric properties were the localization and meticulous of properties of spaces. For the research of pointwise geometric properties, in 1999, Ji donghai first introduced pointwise geometric constants, began to research direction of pointwise geometric contants firstly, and provided some good results. In the paper, we give the notion of the pointwise convex modulus, the relation between the pointwise convex modulus and some geometric propertries are discussed. First of all, we present the development background and process of covex theories and geometric properties of Banach spaces, the significance of pointwise geometric constants and geometric constants of spaces, and we provide discu-ssion of content of each part of the paper. In the second chapter, we introduce the definition of pointwise convex modulus. The relation between numbers of pointwise convex modulus and relevant geometric properties will be discussed. we research the characteristics of numbers of pointwise convex modulus . In the third chapter, in some classical spaces, the estimations and calculatio-ns of pointwise convex modulus are given. The same time, the approximate calculation of pointwise convex modulus in some concrete spaces are given, and we prospect the research contents in the future.