Problems Related To Orthogonalities In Normed Linear Spaces

In this paper, the properties of the entire space are deduced from the properties of generalized orthogonalities and the local point-wise properties of curves related to generalized orthogonalities. It is proved that in any real two-dimensional normed linear space X , there must exist x , y∈X, such that x⊥Iyand x⊥Ey, and then, every central symmetric closed convex curve admits an inscribed square. In addition, Radon curves and Minkowski planes are studied in this paper. It is proved that a Minkowski plane withπ/2-property, whose unit circle is Radon curve, is an inner product space.A large number of significant results have been obtained by several researchers during their studying the relationships between different kinds of generalized orthogonalities as well as between orthogonalities and properties of the spaces. However, these researches mainly focus on the properties of orthogonalities and their impacts on the properties of the entire space, while neglecting the influence which the properties of generalized orthogonalities at some special points of the underlying space would make on the entire space. And few were done on how the point-wise properties of generalized orthogonalities would impact on the entire space.In the second chapter, we obtain a sufficient condition for that there exist two elements x and y which satisfy x⊥Iyand x⊥By. As a result, it is proved that there exist elements x and y with x⊥Iyand x⊥Eyin any real two-dimensional normed linear space. Moreover, it is proved by means of a totally different method that every central symmetric closed convex curve admits an inscribed square. Our result not only focuses on the existence of inscribed square, but also provides more information about the relationship between the inscribed square and the Minkowski plane generated by the curve in which the square inscribes. In the third chapter, it is proved that a Minkowski plane withπ/2-property, whose unit circle is Radon curve, is an inner product space. The results in the part supplement the related research.