Nonsquare Constant In Some Orlicz Sequence Spaces

There is further progress from"qualitative"to"quantitative"of geometric constant to geometric property, which the values range of the geometric constant of space directly decides the naught or exist of some geometric property. The research of geometric constant of space will provide helpful conditions for the research of geometric property. In 1991, J. Gao and K. S. Lau gave two kinds of nonsquare in the sense of James and Sch?ffer, proved that C_J( X ) < 2 is equivalent to that X is uniformly nonsquare and proved that if dim( X )≥2 then 1≤C_S ( X)≤C_J( X)≤2 and C_J ( X)C_S( X) =2 and that C_J( X ) < 3/ 2 implies that the space X have uniformly normal structure. It is well known that uniformly normal structure implies fixed point property, so the research of nonsquare constant has importantly theoretical worthiness. Pointwise geometric constant is a quantitation of pointwise geometric property and local representation of geometric constant of space. The research of pointwise geometric constant includes chiefly its representation, estimation and computation, ect.The geometric theory of Orlicz spaces has been an important part of the geometric theory of Banach space. Recently, the research of nonsquare constant and pointwise nonsquare constant of Orlicz space has been one of topics concerned by a lot of researchers in this area and great advances have been made. The aim of this paper is to discuss the relevant questions about nonsquare constant and pointwise nonsquare constant in Orlicz spaces. For readability and integrity, we firstly make a detailed statement for the progress of kinds of nonsquareness, nonsquare constantand piontwise nonsquare constant and their relevant questions, especially in Orlicz spaces. In the second chapter, representation of some fundamental notations and fundamental results are given and the fundamental theory and the main research results on nonsquare constant...