Isosceles Triangles In Minkowski Planes And Characterizations Of Inner Product Spaces

Properties of Minkowski spaces play an important role in the research of normed linear spaces, and are studied intensively during the last decade. First we survey main existing results in the field of Minkowski Geometry, especially these concerning orthogonality types.Then elementary definitions of linear spaces, distances in linear spaces, isosceles orthogonality, the relation between isosceles orthogonality and other two orthogonality types, namely, Birkhoff orthogonlity and Singer orthogonality, and characterizations of inner product spaces, are collected.We try to extend the property that"the median and altitude on the base of an isosceles triangle coincide"in the Euclidean plane to general Minkowski planes. The concept of altitude of isosceles triangles are introduced, and it is proved that a Minkowski plane is the Eucildean plane if and only if the circumcenter of arbitrary isosceles triangle lies on the line determined by the median of the base of the triangle.