On Chordal Orthogonality In Minkowski Planes

In 2010, Martini and Spirova introduced the notion of chordal orthogonality to Minkowski planes (i.e., to real, two-dimensional normed planes). Based on their result, we study from two different viewpoints the existence of chordal orthogonality, which was previously overlooked; discuss the uniqueness of chordal orthogonality, strengthen Martini and Spirova's result on right-uniqueness of chordal orthogonality, obtain a new result concerning the left-uniqueness of chordal orthogonality and two new characterization of strictly convex Minkowski planes; present the relation of chordal orthogonality to other orthogonality types which were not mentioned in Martini and Spirova's paper. More precisely, the main results of this dissertation includes the following three parts:Existence properties of chordal orthogonality. It is proved that, for a given chord [p1,q1] of the unit circle and a given point p2 on the unit circle, there exists a chord [p2,q2] such that [p1,q1,] is orthogonal to [p2,q2]. A necessary and sufficient condition is provided for the existence of a chord [p1,q1] passing through a given point p1 and orthogonal to a given chord [p2,q2]. It is also shown that, for each chord [p,q] of the unit circle and each numberα∈(0,2], there exists two chords [p1,q1] and [p2,q2] of the unit circle having length a such that [p,q] is orthogonal to [p1,q1] and [p2,q2]is orthogonal to [p,q]. These results complement the study on existence of chordal orthogonality.Relations of the uniqueness of chordal to corresponding properties of the underlying plane. It is proved that chordal orthogonality is right unique if and only if the underlying plane is strictly convex (Martini and Spirova only proved the "sufficiency" part of this result), it is left unique if and only if the underlying plane is both strictly convex and smooth. It is shown that, a Minkowski plane is strictly convex if and only if, for each numberα∈(0,2], any chord [p1,q1,] of the unit circle orthogonal to precisely two chords of the unit circle having length a. Another characterization of strictly convex Minkowski is also presented: for three distinct points on the unit circle satisfying z≠-x and z≠-y, if [x,z] is orthogonal to [z,y] or [z,y] is orthogonal to [x,z], then x≠-y. H. Martini and M. Spirova only showed the sufficiency of this result.The relation of chordal orthogonality to other types of generalized orthogonality types. It is shown that, if chordal orthogonality implies one of following orthogonality types, namely, isosceles orthogonality, Pythagorean orthogonality, Roberts orthogonality, area orthogonality, and Singer orthogonality, then the underlying plane is Euclidean.