The Relation Between Orthogonality Types And The Property Of The Unit Sphere

The unit ball of a normed linear space decides the norm of the underlying space,and therefore decides the metrical property and properties of orthogonality types ofthe underlying space. Conversely, previous work shows that the property of Birkhofforthogonality and isosceles orthogonality can also decide the property of theunderlying space. In view of this, the present thesis clarifies the impact of theexistence of a unit vector having special property with respect to certainorthogonality type on the property of the whole space.First we recall definitions and basic properties of several orthogonality types,some concepts closely related to them, as well as the relation between them, whichlays a solid foundation of the discussion in the sequel.As the first part of our main result we present characterization of isometricreflection vectors in two dimensional normed linear space. Based on this, we provedgeometric characterization of isometric reflections vectors and homogeneousdirections of isosceles orthogonality. The relation between these two notions is alsodiscussed.Then we obtain geometric characterizations ofL2-summand vectors andhomogeneous direction of Pythagorean orthogonality, respectively and show theequivalence of these two notions.Finally, we provides geometric characterizations of I-vector, IP-vector, andP-vector respectively.