Problems Related To Structure Of Bisectors And Inscribed Square In Normed Spaces

Because a real normed linear space does not have a concept of orthogonality similar to the one in the inner product spaces, some geometric objects and geometry problems become extremely complicated. In this paper two geometric problems, which are closely related to concept of generalized orthogonality in normed linear spaces are studied.As one of the main results in this paper, we prove that the bisector of two points is path connected in a real normed linear space with dimension not less than 2. This result improved the result given by Horvath in 2000:bisectors are connected in real normed linear spaces.Another main result in this paper considers one of the hardest open problems in the Euclidean plane, that is, the generalization of problems about inscribed Minkowskian square of simple closed curve in normed planes. We show that any convex Jordan curve in a symmetric Minkowski plane admits an inscribed Singer orthogonality Minkowskian square, and also admits an inscribed area orthogonality Minkowskian square.