Raymond W. Freese

Metrizations of orthogonality and characterizations of inner product spaces

Characterizations of real inner product spaces among normed linear spaces have been obtained by exploring properties of and relationships between various orthogonality relations which can be defined in such spaces. In the present paper the authors present metrized versions of some of these properties and relationships and obtain new characterizations of real inner product spaces among complete, convex, externally convex metric spaces.

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

Abstract A new orthogonality relation in normed linear spaces which generalizes pythagorean orthogonality and isosceles orthogonality is defined, and it is shown that the new orthogonality is homogeneous (additive) if and only if the space is a real inner-product space.

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

A new class of four-point properties which characterize euclidean spaces

Characterizations of generalized euclidean spaces by means of euclidean four-point properties.state that every metric space which is complete, and which contains a metric line joining each two of its points is a generalized euclidean space if and only if each quadruple from a certain class of quadruples of the space is congruent with a quadruple of points in a euclidean space. It is known that it suffices to consider only quadruples containing a linear triple, or quadruples in which one of the linear points is a metric midpoint of the other two. Another class of four-point properties involves quadruples which contain a linear triple and a point equidistant from two of the linear points. The present paper presents three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.

The CMP and Banach Spaces with unique metric lines

This paper is concerned with characterizing the class of Banach Spaces with unique metric lines in the class of complete metric spaces with unique lines.The concept utilized to prove the major theorem is theConsistent Midpoint Property (CMP). We define a binary operation (+) in metric spaces with unique lines and show that, under suitable assumptions, the space is a Banach Space with + as the vector addition and ∥ a∥=d(a, θ) for some fixed θ.

A 2-metric characterization of the euclidean plane

It is known that the euclidean plane can be characterized metrically among the class of all metric spaces by each of several different sets of conditions. In particular, it is shown in [2] that a metric space is congruent with the euclidean plane if and only if it is complete, metrically convex, externally convex and has the property that each 4 of its points are congruently imbeddable in the euclidean plane, with some quadruple non-collinear. A similar question can be considered in a 2-metric (area metric) space ([4-6]) and the purpose of this paper is to prove that under suitable conditions of completeness, convexity and imbeddability of finite sets of points, a 2-metric space may be placed into a one-to-one, area preserving correspondence with the euclidean plane.

Altitude properties and characterizations of inner product spaces

New characterizations of real inner product spaces (euclidean spaces) among metric spaces are obtained from familiar formulas expressing the altitude (height) of a triangle as a function of the lengths of its sides. Other properties related to the altitude of a triangle are also shown to result in characterizations of euclidean spaces, or euclidean and hyperbolic spaces.

Weak additivity of metric pythagorean orthogonality

It is well known that the property of additivity of pythagorean orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes real inner product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.

Strong additivity of metric isosceles orthogonality

It is known that the property of additivity of isosceles orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper it is shown that suitably metrized concepts of additivity of metric isosceles orthogonality characterize euclidean or hyperbolic spaces among complete, convex, externally convex metric spaces.

Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.