Orthogonal Related Issues Generalized Minkowski Space

In this parer, characterizations of inner product spaces are obtained by studyingpointwise properties of Pythagorean orthogonality in normed linear space, a quan-titative of the diference between Birkhof orthogonality and Pythagorean orthogo-nality,is given the existent of biorthogonality in Minkowski space is also discussed.A number of important conclusions results have been obtained by several re-searchers during their studying of properties of generalized orthogonalities of Minkows-ki space and the relationships between diferent kinds of generalized orthogonalities.but, their researches mainly focus on the following felds: the impact and the efecton the properties of Minkowski space by the properties of generalized orthogonalityin the entire space；on the other hand, the research of the relationship between thegeneralized orthogonalities is often qualitative. They mainly focus on two orthogo-nal if there is a diferent on this issue.And lack of quantitative characterizations ofthe real diferences.Based on all that have mentioned above, we frst proved that, We prove thata Minkowski plane X is an inner space if and only if there exists a non-zero P-orthogonally homogeneous element in X. As well as we show that in a Minkowskispace X, if some generalized orthogonality with existence and homogeneity impliesP-orthogonality, then X is an inner product space.Secondly, we depict intuitively the diference between Pythagorean orthogonaland Birkhof orthogonal in this article. From the perspective of quantitative, wedefne a constant P (X) with clear geometric properties. Through calculation, itproved that the scope of the P (X) and the contact with convexity in the spacewhen arrived upper bound. Also, we calculate the exact value of P (X) in theX=lp2space.