Problems Related To Orthogonalities Of The Chords In Normed Linear Space

In this paper, the global property of the space are deduced from the property of generalized orthogonalities and the local properties of chords and related to generalized orthogonalities.It is proved that a Minkowski plane X is Euclidean if and only if the product of the length of two groups of relative chords that Birkhoff arthogona lizes with each other is equivalent, and then, the theorem of two concurrent chords is proved. In addition, circular arcs and Minkowski plane are studied in this paper.It is proved that Minkowski plane is an inner product space if the sums of the length of two groups of relative arcs that Birkhoff orthogonalizes with each other in the unit circle of the Minkowski plane is equivalent.A Minkowski geometry is a fascinating mathematical discipline which is closely related to other mathematical fields, such as convex Geometry and Distance Geometry. A geometric property of the respective Minkowski plane is the circle since, by definition, the shape of the unit circle of a Minkowski plane determines the norm. So it is clear, the more we know about the unit circle, the better we know the geonetric structure of the respective Minkowski plane. On the other hand, since the Euclidean plane is a special Minkowski plane, we always would like to know what extent its properties remain volid for general normed planes, or by what properties they are replaced then.A Minkowski plane X is Eaclidean if and only if the product of the length of two groups of relative chord that Birkhoff orthogonalizes with each other is equivalent, and then, the theorem of two con current chords is proved. Our result not only focuses on the orthogonal chords, but else parallel chords. It is proved from the properties of generalized orthogonalities and the properties of the space that Minkowski plane is an inner product space if the sums of the length of two groups of relative arcs that Birkhoff arthogonalizes with each other in the unit circle of the Minkowski plane is equivalent.The results is the part supplement for the related research.