Calculation And Representation Of Nonsquare Constants In Symmetric Minkowski Plane

A symmetric Minkowski plane is a real two-dimensional Banach space with symmetric bases. Some examples of symmetric Minkowski planes are l p2, and spaces whose unit sphere are regular 2n -polygons(n is even). Therefore, it is very important in theory to study geometric constants, such as non-square constants, in symmetric Minkowski planes.In this paper, by making use of characteristics of isosceles orthogonality, non-square constants in symmetric Minkowski planes are studied. The paper is divided into three parts and is organized as follows:In Chapter 1, the development of the geometry of Banach space, especially the study of non-square constants, is presented.In Chapter 2, basic definitions and conclusions concerning non-square constants, isosceles orthogonality and symmetric Minkowski planes are presented. It is proved that there are at least two pairs of symmetric axes in a symmetric Minkowski plane. It is also proved that infinitely many symmetric Minkowski planes can be constructed based on two given symmetric Minkowski planes, which shows the extensive existence of symmetric Minkowski planes.Besides, some properties of generalized non-square constant C J( a ,X ) introduced by S. Dhompongsa are studied. A equivalent representation of generalized non-square constant, which improves the condition"x , y ,z∈B ( X) of which at least one belongs to S ( X )"to"y ,z∈B ( X) and x∈S ( X)", is presented. As a result, it is proved that C J( a , X )= 2 if a≥2. Therefore, we just need to study C J( a ,X ) when a∈[ 0,2).