Research On Orthogonality And Problems Related To Circles In Minkowski Spaces

There are many perfect results on orthogonality in Euclidean geometry, which are useful in the study of problems about Euclidean spaces. With the development of Minkowski geometry (i.e., geometry of finite dimentional normed linear spaces), many scholars introduce and study carefully different kinds of generalized orthogonalities in Minkowski spaces (i.e., finite dimentional normed linear spaces). However, properties of these generalized orthogonalities, especially some of basic properties, deserve to be solved. In this paper, we study the uniqueness of isosceles orthogonality in Minkowski spaces and give a best result in this direction.First, we collect main results in this research field and show the objective and main problems of this paper. Second, definitions and basic properties of some concepts in Minkowski spaces are collected, definitions and main properties of isosceles orthogonality and some known results concerning orthogonality are presented.Finally, by studying the relation between isosceles orthogonality and the lengths of segments contained in the unit sphere, it is proved that for any point x in a Minkowski plane X satisfying |x| > 0and for each numberρ∈[0,2|x|/M_x] (ρ∈[ 0,+∞) when M_x= 0), there exists a unique point y∈ρS_X (except for the sign) such that x⊥Iy. This result improves the result obtained by J. Alonso in 1994, which says that for any point x a Minkowski plane X and for each number 0 <ρ≤| x|, there exists a unique point y∈ρS_X (except for the sign) such that x⊥Iy."...