Finite Dimensional Banach Spaces Research Articles

Finite-dimensional Banach spaces with numerical index zero

We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X 0 and nonzero complex spaces X 1 ,...,X n in such a way that the equality ∥x 0 + e iq1ρ x 1 +...+e iqnρ x n ∥ = ∥x 0 +...+ x n ∥ holds for suitable positive integers q 1 ,...,q n , and every ρ ∈ R and every x j ∈ X j (j = 0, 1,...,n). If the dimension of X is two, then the above result gives X = C, whereas dim(X) = 3 implies that X is an absolute sum of R and C. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.

PLANK PROBLEMS, POLARIZATION AND CHEBYSHEV CONSTANTS

In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical <TEX>$L^{p}(\mu)$</TEX> spaces. In the case <TEX>$1\;{\leq}\;p\;{\leq}\;2$</TEX> we obtain optimal results and for finite dimensional complex Banach spaces, in a special case, we have improved an early result by K. Ball [3]. By using these results, in some cases we are able to find best possible lower bounds for the norms of homogeneous polynomials which are products of linear forms. In particular, we give an estimate in the case of a real Hilbert space which seems to be a difficult problem. We have also obtained some results on the so-called n-th (linear) polarization constant of a Banach space which is an isometric property of the space. Finally, known polynomial inequalities have been derived as simple consequences of various results related to plank problems.

The geometry of minkowski spaces — A survey. Part II

In this second part of a series of surveys on the geometry of finite dimensional Banach spaces (Minkowski spaces) we discuss results that refer to the following three topics: bodies of constant Minkowski width, generalized convexity notions that are important for Minkowski spaces, and bisectors as well as Voronoi diagrams in Minkowski spaces.

The Steiner ratio of high-dimensional Banach–Minkowski spaces

The Steiner ratio is the greatest lower bound of the ratios of the Steiner Minimal Tree- by the Minimum Spanning Tree-lengths running over all finite subsets of a metric space. We will discuss this quantity for finite-dimensional Banach spaces of high dimension. Particularly, let the quantity C d defined as the upper bound of the Steiner ratio of all d-dimensional Banach spaces, then lim d → ∞ C d = lim d → ∞ m d ( B 2 ) ) , where m d ( B(2)) denotes the Steiner ratio of the d-dimensional Euclidean space.

The existence of nonzero solutions for a class of variational inequalities by index

The existence of nonzero solutions for a class of variational inequalities is studied by a fixed-point index approach for multivalued mappings in finite-dimensional spaces and reflexive Banach spaces.

Numerical-Radius-Attaining Polynomials

In this paper we discuss a version of James theorem making use of polynomials which attain their numerical radius. In addition, we obtain a characterization of finite-dimensional Banach spaces in terms of such polynomials.

Contractive projections and operator spaces

Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces H n k H_n^k , 1 ≤ k ≤ n 1\le k\le n , generalizing the row and column Hilbert spaces R n R_n and C n C_n , and we show that an atomic subspace X ⊂ B ( H ) X\subset B(H) that is the range of a contractive projection on B ( H ) B(H) is isometrically completely contractive to an ℓ ∞ \ell ^\infty -sum of the H n k H_n^k and Cartan factors of types 1 to 4. In particular, for finite-dimensional X X , this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w ∗ ^* -closed J W ∗ JW^* -triples without an infinite-dimensional rank 1 w ∗ ^* -closed ideal.

Embedding Levy families into Banach spaces

We prove that if a metric probability space with a usual concentration property embeds into a finite dimensional Banach space X, then X has a Euclidean subspace of a proportional dimension. In particular this yields a new characterization of weak cotype 2. We also find optimal lower estimates on embeddings of metric spaces with concentration properties into $ l^k_\infty $ , generalizing estimates of Bourgain—Lindenstrauss—Milman, Carl—Pajor and Gluskin.

OPERATORS FROM CERTAIN BANACH SPACES TO BANACH SPACES OF COTYPE q ≥ 2

Suppose { <TEX>$X_{n}$</TEX>}<TEX>$_{n=1}$</TEX><TEX>$^{\infty}$</TEX> sequence of finite dimensional Banach spaces and suppose that X is either a closed subspace of (equation omitted) or a closed subspace of (equation omitted) with p>2. We show that every bounded linear operator from X to a Banach space Y of cotype q(2<TEX>$\leq$</TEX>q〈p) is compact.t.t.

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.

The vertex degrees of minimum spanning trees

We study the problem of minimum spanning trees (MST) with degree constraints. It is well-known that this problem in general is NP -hard. It will be shown that in finite-dimensional Banach spaces there is a number such that a bounded degree MST can be computed as efficiently as an ordinary MST if the degree constraint is greater than this number.

Generalized Stability of Isometries

Let X and Y be real Banach spaces and let ε,p≥0. A mapping f: X→Y is called an (ε,p)-isometry if |‖f(x)−f(y)‖−‖x−y‖|≤ε‖x−y‖p holds for all x,y∈X. A pair (X,Y) is p-stable with respect to isometries if there exists a function δ: [0,∞)→[0,∞) with limε→0δ(ε)=0 such that for every surjective (ε,p)-isometry f: X→Y there is a surjective isometry U: X→Y satisfying the estimate ‖f(x)−U(x)‖≤δ(ε)‖x‖p, x∈X. We show that every pair of Banach spaces (X,Y) is p-stable for 0≤p<1. The pair (R2,R2) is not 1-stable. When p>1 a superstability phenomenon occurs for finite-dimensional Banach spaces.

Almost locally minimal projections in finite dimensional Banach spaces

A projectionP on a Banach spaceX is called “almost locally minimal” if, for every α>0 small enough, the ballB(P,α) in the spaceL(X) of all operators onX contains no projectionQ with\(\left\| Q \right\| \leqslant \left\| P \right\|\left( {1 - D\alpha ^2 } \right)\) whereD is a constant. A necessary and sufficient condition forP to be almost locally minimal is proved in the case of finite dimensional spaces. This criterion is used to describe almost locally minimal projections on l1n.

Local stability of the Cauchy and Jensen equations in function spaces

Let X and Y be finite dimensional Banach spaces and let \( D\subset X \) be a convex set such that \( 0\in D \). We prove that the Cauchy equation is stable in the Lipschitz norm for functions defined on D and taking values in Y. We consider also the Jensen equation.

On Benson’s Definition of Area in Minkowski Space

AbstractLet (X, ‖ . ‖) be a Minkowski space (finite dimensional Banach space) with unit ball B. Various definitions of surface area are possible in X. Here we explore the one given by Benson [1], [2]. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.

Existence Theorems of Hartman–Stampacchia Type for Hemivariational Inequalities and Applications

We give some versions of theorems of Hartman-Stampacchia‘s type for the case of Hemivariational Inequalities on compact or on closed and convex subsets in infinite and finite dimensional Banach spaces. Several problems from Nonsmooth Mechanics are solved with these abstract results.

Convexity and Openness with Linear Rate

This paper presents conditions for openness with linear rate (or, equivalently, for metric regularity) of continuous mappings that possess certain convexity properties.Convex continuous functions on a Banach space are proved to be open with linear rate around each point that is not a minimum point. For continuous mappings that are convex with respect to a normal cone in a finite dimensional Banach space as image space, a sufficient condition for openness with linear rate is given. Special cases are treated: For Fréchet-differentiable cone–convex mappings, the surjectivity of the derivative is proved to be equivalent to openness with linear rate. Finitely generated cones lead to a sufficient condition for openness with linear rate that simplifies practical use.A tangency formula of Lyusternik-type is set up for mappings that are open with linear rate, and is applied to cone–convex mappings.

Some Characterizations of Finite-Dimensional Hilbert Spaces

LetXbe a finite-dimensional Banach space. The following affirmations are equivalent:•is a Hilbert space.•Every extreme operator inis an isometry.•The unit ball of(), the space of linear and continuous operators in, is the convex hull of its isometries.•The unit ball of() is the closed convex hull of its isometries.

BOUNDEDNESS STABILITY PROPERTIES OF LINEAR AND AFFINE OPERATORS

Let $E$ be a vector space in which some notion of boundedness is defined. Then $T:E\to E$ is said to have the boundedness stability property (BSP) if for each $x\in E$, the sequence $(T^n_x)^\infty_{n=1}$ is bounded whenever a subsequence $(T^{n_i}x)^\infty_{i=1}$ is bounded. It is shown that (1) every affine operator on a finite-dimensional Banach space has the (BSP); (2) every affine operator on an infinite-dimensional vector space has the functional (BSP); (3) when $E$ is an infinite-dimensional Banach space, an affine operator $T$ on $E$ has the (BSP) if its linear part $A_T=T-T(0)$ is a compact perturbation of a bounded linear operator with spectral radius less than one and (4) when $E$ is a Hilbert space, every normal or subnormal bounded linear operator has the (BSP). Some results on affine operators on a Hilbert space whose linear parts are normal or subnormal are also obtained. Finally, some problems are posed.

Subspaces of L p Isometric to Subspaces of ℓ p

We present three results on isometric embeddings of a (closed, linear) subspace X of Lp=Lp[0,1] into lp . First we show that if p ∉ 2N, then X is isometrically isomorphic to a subspace of lp if and only if some, equivalently every, subspace of Lp which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p ∈ 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of lp, where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of Lp .