Four-point characterizations of real inner product spaces

Generalized euclidean spaces have been characterized among metric spaces by the requirement that each member of certain classes of quadruples of points of the metric space be congruent to a quadruple of points of a euclidean space. The present paper strengthens earlier characterizations which only require the embedding of certain classes of quadruples which contain a linear triple and in which some three of the six distances between pairs of points are equal. These results generalize some similar characterizations of euclidean spaces among normed linear spaces. Received 4 January 1999; revised 12 August 2002.     

An equilateral four point property which characterizes generalized euclidean spaces

Various characterizations of generalized euclidean spaces among complete metric spaces which contain a metric line joining each two points make use of euclidean four point properties, which require that every quadruple from a suitably chosen class of quadruples of points of the metric space be isometric with a quadruple of euclidean points. The present paper shows if every quadruple which contains an equilateral triple and a linear triple is embeddable, the space is generalized euclidean.     

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.     

Angles of Norm ed Linear Spaces and Characterizations of Inner Product Spaces

§１．　Ｄｅｆｉｎｉｔｉｏｎｓ，ＬｅｍｍａｓａｎｄＴｈｅｏｒｅｍＤｅｆｉｎｉｔｉｏｎ１［１］　Ｌｅｔ（Ｘ，ｙ·ｙ）ｂｅａｒｅａｌｎｏｒｍｅｄｌｉｎｅａｒｓｐａｃｅｏｆｄｅｍｅｎｓｉｏｎｇｒｅａｔｅｒｔｈａｎ１，ｗｅｄｅｆｉｎｅａｎｇｌｅｓＡｍ，ｎ（ｘ，ｙ）ｂｅｔｗｅｅｎｎｏｎｚｅｒｏｖｅｃｔｏｒｓｘａｎｄｙｉｎＸｂｙＡｍ，ｎ（ｘ，ｙ）＝ｃｏｓ－１ｎｙｘｙｘｙ＋ｙｙｙｙｙ２－ｍｙｘｙｘｙ－ｙｙｙｙｙ２＋２（ｍ－ｎ）２（ｍ＋ｎ），ｈｅｒｅｍａｎｄｎａｒｅｔｗｏｎｏｎ－ｎｅｇｅｎｔｉｖｅｉｎｔｅ－ｇｅ…     

Flatness of bisectors and the symmetry of distance

H. Busemann proved that the elementary spaces are characterized by the flatness of bisectors among all G-spaces with symmetric distance. In this paper we prove that in two dimensional straight G-spaces with a not necessarily symmetric distance the symmetry of distance is in fact implied by the flatness of bisectors. Thus Busemann's characterizations of the euclidean and hyperbolic planes are shown to hold among all straight two dimensional G-spaces with a not necessarily symmetric distance.     

Some remarkable lines of triangles in real normed spaces and characterizations of inner product structures

Summary In this work we consider the heights and the bisectrices of a triangle in a real normed space. Using well-known formulas which can be generalized to real normed spaces we obtain a collection of new characterizations of inner product spaces.     

CHARACTERIZATIONS OF COMPACT OPERATORS ON SOME EULER SPACES OF DIFFERENCE SEQUENCES OF ORDER m

In the past, several authors studied spaces of m-th order difference sequences, among them, H.Polat and F.Basar ([17]) defined the Euler spaces of m-th order difference sequences e r 0 (△ ( m ) ), e r c (△ ( m ) ) and e r ∞ (△ ( m ) ) and characterized some classes of matrix transformations on them. In our paper, we add a new supplementary aspect to their research by characterizing classes of compact operators on those spaces. For that purpose, the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0 , c and ∞ . The main tool for our characterizations is the Hausdorff measure of noncompactness.     

Geometric Properties Characteristic of Normed Linear Space

This paper provides new characterizations of arbitrary, not necessarily rotund, normed linear spaces, using triangle median properties and weakened forms of associativity and the consistent midpoint property. Connections are explored between the methods used here and a characterization announced by ARONSZAJN in 1935.     

The geometry of Minkowski spaces — A survey. Part I

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons.     

Generalized Cauchy functional equation and characterizations of inner product spaces

Summary Using a generalized Cauchy functional equation we show that some well-known characterizations of inner product spaces, such as those of Jordan—von Neumann, Johnson, and Rassias, can be proved without use of the triangle inequality.     

Relationship Between Quantum and Poisson Structures of Odd Dimensional Euclidean Spaces

It is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are homeomorphic to the Poisson prime and Poisson primitive spectra of the multiparameter Poisson algebra of odd-dimensional euclidean spaces in the case when the multiplicative subgroup of a base field generated by the parameters is torsion free. As a corollary, it is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring.     

Alcune osservazioni riguardanti i gruppi di LieG 2 e Spin(7), candidati a gruppi di olonomia

Summary The Lie groups G₂ and Spin(7) can be considered as automorphisms groups of euclidean vector spaces (of dimension 7, 8 resp.) endowed with a suitable vector product (cfr. [12]). Here one put in evidence several geometric properties of certain special subspaces of such euclidean spaces and the manifolds of special subspaces are determined as well known homogeneous spaces. One considers also riemannian manifolds with holonomy group G₂ or Spin(7) establishing that in the analytic case the existence of a totally geodesic submanifold of codimension 1 imply local reducibility.

---

---

Lavoro svolto nell'ambito del «Gruppo Nazionale Structure Algebriche, Geometriche e Applicazioni (Consiglio Nazionale delle Ricerche)».     

Wackelige Gelenkwerke und wackelige Polyeder in Der sphärischen Geometrie

Infinitesimal rigidity and shakiness of jointed rodworks or polyhedra have been of major interest in recent research in geometry. In this paper structures of this type are investigated in spheres and projective spaces. A transformation is given showing that essentially no new structures occur in these spaces compared with the euclidean case. The main tool is the projective invariance of shaky structures. Shaky structures in euclidean space can be obtained as affine sections of shaky structures in a projective space of the same dimension.     

Some New Results Related to Compact Matrix Operators in the Class((l_p)T,l_∞)

The main goal of this paper is to establish necessary and sufficient conditions for a matrix A ∈((lp)T,l∞), where T is an arbitrary triangle, 1 ≤ p ≤∞, to be a compact operator. In the past,only sufficient conditions were established in almost all of those cases, by using the Hausdorff measure of noncompactness. We improve those results by applying another method for the characterizations of compact linear operators between BK spaces.     

Kirszbraun's Theorem and Metric Spaces of Bounded Curvature

We generalize Kirszbraun's extension theorem for Lipschitz maps between (subsets of) euclidean spaces to metric spaces with upper or lower curvature bounds in the sense of A.D. Alexandrov. As a by-product we develop new tools in the theory of tangent cones of these spaces and obtain new characterization results which may be of independent interest. Submitted: June 1996, final version: November 1996     

Generalization of tangential trace spaces of H(curl,Ω) for curvilinear Lipschitz polyhedral domains Ω

For curvilinear Lipschitz polyhedral domains Ω, explicit characterizations of the tangential trace spaces of H ¹(Ω) are presented. These extend the original characterizations given by Buffa and Ciarlet that hold on Lipschitz polyhedral domains with plane faces. The tangential trace spaces of H ¹(Ω) are fundamental for the definition, analysis and intuitive understanding of the trace spaces of H ( curl ,Ω) and therefore, more general characterizations of the latter are obtained at the same time. Copyright © 2013 John Wiley & Sons, Ltd.     

Giuseppe Tallini: His life and work

It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of real inner product spaces among complete, convex, externally convex metric spaces is obtained.     

Analytic and Luzin spaces (non-separable case)

We introduce the concept of κ-analytic and κ-Luzin spaces as images of complete metric spaces by (disjoint) upper semi-continuous compact-valued correspondences which “preserve discreteness” in some sence (Definition in Section 3.1). The case κ = ω coincides with (Lindelöf) analytic spaces studied by Choquet, the first author and others. The main results are characterizations of uniform analytic spaces in terms of other parametrizations, complete sequences of covers, and Suslin subsets of some product of a compact and a complete metric space (Theorems in Section 3.2 and in Section 4), and characterizations of topological analytic spaces as Suslin subsets of paracompact ?ech-complete spaces (Theorem in Section 5).     

The characterizations of triangles using the nine-point circles

In this note, primarily intended for high school students and high school teachers, characterizations of a right triangle and an equilateral triangle in the Euclidean plane are presented using the nine-point circle of a given triangle. Geometrical applications are explored along with their possible uses in the teaching environment.     

Spectral Theory of Riesz Potentials on Quasi‐Metric Spaces

This paper deals with spectral assertions of Riesz potentials in some classes of quasi‐metric spaces. In addition we survey briefly a few related subjects: integral operators, local means and function spaces, euclidean charts of quasi‐metric spaces, relations to fractal geometry.