We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations. Mathematics Subject Classification (2000). 46B20, 46C15, 52A21.

In this paper we introduce a new geometry constant D(X) to give a quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality. We show that 1 and 2( √ 2 − 1) is the upper and lower bound for D(X), respectively, and characterize the spaces of which D(X) attains the upper and lower bounds. We calculate D(X) when X = (R 2 , · p ) and when X is a symmetric Minkowski plane respectively, we show that when X is a symmetric Minkowski plane D(X) = D(X * ).

Via studying the relation between isosceles orthogonality and the lengths of segments contained in the unit sphere, existing results on the uniqueness of isosceles orthogonality are improved.Mathematics Subject Classification (2010). Primary 46B20.

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It is well known that the construction of Voronoi diagrams is based on the notion of bisector of two given points. Already in normed linear spaces, bisectors have a complicated structure and can, for many classes of norms, only be described with the help of topological methods. Even more general, we present results on bisectors for convex distance functions (gauges). Let C, with the origin o from its interior, be the compact, convex set inducing a convex distance function (gauge) in the plane, and let B(−x, x) be the bisector of −x and x, i.e., the set of points z such that the distance (measured with the convex distance function induced by C) from z to −x equals that from z to x. For example, we prove the following characterization of the Euclidean norm within the family of all convex distance functions: if the set L of points x in the boundary ∂C of C that creates B(−x, x) as a straight line has non-empty interior with respect to ∂C, then C is an ellipse centered at the origin. For the subcase of normed planes we give an easier approach, extending the result also to higher dimensions.