Some Problems Of Nonsquare Constants In Normed Spaces

In 1990,J.Gao and Lau Ka-Sing introduced James nonsquare constants and Schaffer constants, and in 1999, Ji Donghai and Wang Shuxin gave the defination of pointwise nonsquare constants.Here we provide further discussion on this subject.The main results of the thesis are summarized as follows: Chapter 1 Introduction: We recall the development process of the theory of Orlicz spaces and the pioneer's main research results during the past 60 years. This paper introduces the research and the significance of the geometric constants and pointwise geometric constants.Then this chaper presents the background and the significance of the contents that we discuss next. Chapter 2 Nonsquare constants of Orlicz spaces: geometric constants of sp-ace is a quantitative index of its corresponding geometric property.From investigating geometric property to compute geometric constant is a progressfrom"qualitative"to"quantitatie". The work of Ren Zhongdao et al. gave emphasis to the approximate calculation of the nonsquare constants in some concrete Orlicz space.Ji Donghai et al. gave emphasis to estimating and showing the nonsqure constants by the supremum and infremum of the elements in unit unit sphere.In this paper,for some Orlicz sequence space we obtain the representation of nonsquare constants by N-function. We also obtain the estimation of nonsquare constants on Orlicz sequence space and functional space endowed with Luxemburg norm on some conditions, the exact value of nonsquare constants is also given.Meanwhile we discuss the nonsquare constants in some classical spaces and the relations between the value of nonsquare constants in space and its dual space. Chapter 3 pointwise nonsquare constants of space:The pointwise property is the localization of the property of space.pointwise geometric constants is the quantitatie of pointwise geometric property of spaces,and is the localization of geometric constants of spaces.Some pointwise geometry properties directly depict the geometry speciality of space.In this paper,we study emphatically two dimension space,the relations of pointwise nonsquare constants and the relations between the value of pointwise nonsquare constants and other geometric properties.