Finite-dimensional Normed Linear Spaces Research Articles

Klee-Phelps convex groupoids

Abstract We prove that a pair of proximal Klee-Phelps convex groupoids A(∘), B(∘) in a finite-dimensional normed linear space E are normed proximal, i.e., A(∘) δ B(∘) if and only if the groupoids are normed proximal. In addition, we prove that the groupoid neighbourhood N z (∘) ⊆ S z (∘) is convex in E if and only if N z (∘) = S z (∘).

Orthogonal bases and a structure of finite dimensional normed linear spaces

In this paper, we consider orthonormal bases for finite dimensional normed linear spaces which are related to Birkhoff orthogonality. We also study the relationship between orthonormal bases and a structure of finite dimensional normed linear spaces.

Finite Dimensional Generating Spaces of Quasi-Norm Family

In this paper, some results on nite dimensional generating spaces of quasi-norm family are established. The idea of equivalent quasi-norm fami- lies is introduced. Riesz lemma is established in this space. Finally, we re-dene B-S fuzzy norm and prove that it induces a generating space of quasi-norm family. Chang et al.(2) dealt with the generating spaces of quasi-metric family and by studying the common characteristic properties of both fuzzy metric spaces in the sense of Kaleva & Seikkala (6) and Menger probabilistic metric spaces(11) estab- lished that the generating spaces of quasi-metric family possess the unification characterization of the fuzzy and probabilistic situations. They also proved sev- eral fixed point theorems in generating spaces of quasi-metric family. Jung et al.(5) established some fixed point theorems in generating spaces of quasi-metric family. Many authors introduced different ideas of fuzzy norm ((9), (10), (13)), fuzzy n-norm (3), fuzzy Hilbert space(4) etc. in different ways. In 2006, Xiao & Zhu(12) introduced a concept of a generating space of quasi-norm family (G.S.Q.N.F) and studied its properties in a linear topological space setting. They also introduced the concept of convergent sequences, Cauchyness, completeness, compactness, etc. and extended some fixed point theorems, especially Schauder-type fixed point theorem in such spaces. In this paper, we study certain results in finite dimensional generating spaces of quasi-norm family (G.S.Q.N.Fs). We observe that most of these results more or less differ from the results in finite dimensional normed linear spaces. We generalize the definition of fuzzy norm as introduced by Bag & Samanta(1) (B-S fuzzy norm), and we deduce a G.S.Q.N.F from it. The organization of the paper is as follows: Section 1 comprises of some preliminary results. In section 2, we study some results in finite dimensional G.S.Q.N.Fs. In section 3, a generalized form of B-S fuzzy norm is advanced and a G.S.Q.N.F is deduced from it. Throughout this paper straightforward proofs are omitted.

Auerbach Bases and Minimal Volume Sufficient Enlargements

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Let(X,d)be a pointed compact metric space, let0<α<1, and letφ:X→Xbe a base point preserving Lipschitz map. We prove that the essential norm of the composition operatorCφinduced by the symbolφon the spaceslip0(X,dα)andLip0(X,dα)is given by the formula‖Cφ‖e=limt→0 sup⁡0<d(x, y)<t(d(φ(x),φ(y))α/d(x,y)α)whenever the dual spacelip0(X,dα)∗has the approximation property. This happens in particular whenXis an infinite compact subset of a finite-dimensional normed linear space.

Concurrent and parallel chords of spheres in normed linear spaces

We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.

Characterizations of the Nonemptiness and Boundedness of Weakly Efficient Solution Sets of Convex Vector Optimization Problems in Real Reflexive Banach Spaces

Under a weak compactness assumption on the functions involved, which always holds in finite-dimensional normed linear spaces, this paper extends various characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems, obtained previously by the author (Deng in J. Optim. Theory Appl. 96:123–131, 1998) in the real finite-dimensional normed linear space setting, to those in the real reflexive Banach space setting.

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let B Y denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P : Y → X such that P ( B Y ) ⊂ A . The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.

Extensions of a Bonnesen-style inequality to Minkowski spaces

Various definitions of surface area and volume are possible in finite dimensional normed linear spaces (= Minkowski spaces). Using a Bonnesen-style inequality, we investigate the ratio of the Holmes-Thompson surface area of the unit ball to its volume. In particular, in the planar case a stronger lower bound for this ratio is established when the area is defined in the sense of Holmes-Thompson, or is given by the definition of mass. From this we obtain some (characteristic) properties of Radon curves. Mathematics subject classification (2000): 52A10, 52A21, 52A40, 46B20.

Strong proximinality and renormings

We characterize finite-dimensional normed linear spaces as strongly proximinal subspaces in all their superspaces. A connection between upper Hausdorff semi-continuity of metric projection and finite dimensionality of subspace is given.

Low-degree minimal spanning trees in normed spaces

We give a complete proof that in any finite-dimensional normed linear space a finite set of points has a minimal spanning tree in which the maximum degree is bounded above by the strict Hadwiger number of the unit ball, i.e., the largest number of unit vectors such that the distance between any two is larger than 1.

On the convexity of the weakly compact Chebyshev sets in Banach spaces

A sufficient condition for a Banach spaceX is given so that every weakly compact Chebyshev subset ofX is convex. For this purpose a class broader than that of smooth Banach spaces is defined. In this way a former result of A. Brondsted and A. L. Brown is partially extended in every finite dimensional normed linear space and a known result in Hilbert spaces is proved in a different way.

Sums of distances in normed spaces

Abstract : A geometric proof for the following theorem due to Martelli and Busenberg is given. Integral geometry is used to discuss special cases and related results. Minkowski spaces are simply finite dimensional normed linear spaces. Smoothness assumptions on the boundary of the unit disk E for a Minkowski plane will enable us to use Crofton's simplest formula from integral geometry to give a proof for three points. If the unit ball for a 3-dimensional Minkowski space is a zonoid, then we used integral geometry for the case of four points forming a simplex. A zonoid is a limit of sums of segments. Bolker discusses equivalent conditions for a convex subset of Rn to be a zonoid.

Measuring flatness in linear spaces

Let L be a finite-dimensional normed linear space and let M be a compact subset of L lying on one side of a hyperplane through 0. A measure of flatness for M is the number D(M) = inf{ sup f(x) f(y) : x, y ϵ M} , where the infimum is over all f in L ∗ which are positive on M. Thus D( M) = 1 if M is flat, but otherwise D( M) > 1. On the other hand, let E( M) be a second measure on M defined as follows: If M is linearly independent, E( M) = 1. If M is linearly dependent, then (1) let Z be a minimal, linearly dependent subset of M; (2) partition Z into mutually exclusive subsets U = { u 1, …, u p } and V = { v 1, …, v q } such that there exist positive coefficients a i and b i for which Σ i = 1 p a i u i = Σ i = 1 q b i v i ; (3) let r = max{ Σ i = 1 p a i Σ i = 1 q b i , Σ i = 1 p b i Σ i = 1 q a i } ; (4) let E( M) be the supremum of all ratios r which can be formed by steps (1), (2) and (3). The main result of this paper is that these two measures are the same: D( M) = E( M). This result is then used to obtain results concerning the Banach distance-coefficient between an arbitrary finite-dimensional normed linear space and Hilbert space.

A representation of the norm in finite-dimensional normed linear spaces

A representation of the norm in finite-dimensional normed linear spaces

On the distance between polytopes

This paper is written to fill an apparent gap in the literature in connection with some elementary problems of distance in finite dimensional normed linear spaces. In particular the problem of determining the distance between two polytopes in a normed n n -dimensional real vector space is considered. Special consideration is given to the case in which the norm is a twice differentiable function of its arguments and for this case a convex programming algorithm is presented. In addition several other cases are considered including the well known discrete Tchebycheff approximation. The results should find application in approximation theory.

A remark on the intersection of shells

There are many characterizations which distinguish finite-dimensional normed linear spaces from infinite-dimensional normed linear spaces. Perhaps the best known of these is the compactness of the unit ball. Recently, V. Klee [1] showed that in any infinite-dimensional normed linear space there exists a decreasing sequence of unbounded but linearly bounded closed convex sets whose intersection is empty. We will give here a somewhat similar condition which holds in all infinite-dimensional normed linear spaces but does not hold in any finite-dimensional space. We begin with the following terminology. U will denote the unit ball {x: ||x|| S 1 } and S the unit sphere {x: ||x|| = 1 }. A shell will be any set of the form {x: ri? IIx-xalI ? r2 for 0 1 such that (x+p U)Cr (-x+p U) CO, and S will be called finitely nonfiat if there exists a finite number of points {xi} '1 in S such that n(xj+pU) Co. A collection of sets has the finite intersection property if the intersection of the sets in any finite subcollection is not empty.