The Related Problems With Orthogonality And Transitivity Of The Norm

Clearly, geometric properties of a normed linear space are fully determined bythe shape of its unit ball. Conversely, it’s a natural question whether some localproperties of a space can determine or in certain way determine the global propertyof the space. In this thesis, we study equipower properties and the homogeneity ofPythagorean orthogonality.As one part of the main results in this thesis, the definition of equipower point isextended to general Minkowski planes by replacing the distance in i~2with thedistance induced by norm，and it is proved that there is no equipower point exceptfor the origin in the interior of the unit circle of l_p~2(1≤p≤∞, p≠2). The possibleregion in which equipower points of the unit circle might exist is presented in a non-strictly convex normed plane via a geometric method. By applying the transitivity ofthe norm, it is proved that if the unit circle of a Minkowski plane admits anequipower point other than the origin as well as a big point, then the space is aHilbert space.In the second part of the main results in this thesis, we consider thehomogeneity of Pythagorean orthogonality in normed linear spaces. It is proved thatPythagorean orthogonality is unique at x if x is a homogeneous direction ofPythagorean orthogonality. At the same time, we study the relation between thehomogeneous direction of Pythagorean orthogonality and other notions includingisometric reflection vectors andL2-summand vectors, and show that a Banach spaceX is a Hilbert space if and only if the relative interior of the set of homogeneousdierections of Pythagorean orthogonality in the unit sphere of X is not empty. Weintroduce a geometric constantNPX to measure the non-homogeneity ofPythagorean orthogonality. It is proved that NP_X=0if and only if X is a Hilbertspace.