The geometry of Minkowski spaces — A survey. Part I

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.

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

Studies in Geometry. By L. M. Blumenthal and K. Menger. Pp. xiv, 512. £7. 1970. (Freeman.)

Stone-type theorem on b-metric spaces and applications

La solution du probleme de valeurs limites mixte tridimensionnel pour le Laplacien dans un domaine polyedrique a des formes singulieres particulieres aux angles et aux bords. Une decomposition du type produit tensoriel de ces formes singulieres le long des bords est effectuee. On presente un systeme fortement elliptique d'equations integrales frontiere qui est equivalent au probleme de valeurs limites mixte. On donne des resultats de regularite pour la solution du systeme qui permettent d'analyser l'influence d'un maillage gradue sur le taux de convergence des solutions d'elements frontiere de Galerkin correspondantes. On montre qu'il suffit de raffiner le reseau maille seulement du cote des bords de la surface pour regagner le taux optimal de convergence

Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes

Angles in Normed Linear Spaces and a Characterization of Real Inner Product Spaces

Angle Bisectors in Normed Linear Spaces

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Existence of 2-segments in 2-metric spaces

An extension of pythagorean and isosceles orthogonality and a characterization of inner-product spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].

Edward Z. Andalafte

Papers

Angles in Normed Linear Spaces and a Characterization of Real Inner Product Spaces

Angle Bisectors in Normed Linear Spaces

Existence of 2-segments in 2-metric spaces

An extension of pythagorean and isosceles orthogonality and a characterization of inner-product spaces

Characterization of inner product spaces in V- and L-spaces