Finite Dimensional Banach Spaces Research Articles

Semi-Inner Products and the Concept of Semi-Polarity

The lack of an inner product structure in Banach spaces yields the motivation to introduce a semi-inner product with a more general axiom system, one missing the requirement for symmetry, unlike the one determing a Hilbert space. We use it on a finite dimensional real Banach space $(\X, \| \cdot\|)$ to define and investigate three concepts. First, we generalize that of \emph{antinorms}, already defined in Minkowski planes, for even dimensional spaces. Second, we introduce \emph{normality maps}, which in turn leads us to the study of \emph{semi-polarity}, a variant of the notion of polarity, which makes use of the underlying semi-inner product.

Constant sign solutions for parameter-dependent superlinear second-order difference equations

This paper studies the existence of constant sign solutions for a second-order parameter-dependent super linear difference equation. The approach is based on variational methods on finite dimensional Banach spaces.

Tangents and Rectifiability of Ahlfors Regular Lipschitz Differentiability Spaces

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger in [Ch99]. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Banach space. In contrast, we also show that if an Ahlfors regular Lipschitz differentiability space has charts of non-maximal dimension, then these charts are strongly unrectifiable in the sense of Ambrosio-Kirchheim.

Discrete boundary value problem based on the fractional Gâteaux derivative

This article aims to organize positive solutions to discrete fractional boundary value problems for continuous Gâteaux differentiable functions. A generalized Gâteaux derivative is introduced using a fractional discrete operator for a Jumarie fractional operator. The method of finding solutions is based on critical point theorems of finite dimensional Banach spaces.

ON THE EXISTENCE OF 1‐SEPARATED SEQUENCES ON THE UNIT BALL OF A FINITE‐DIMENSIONAL BANACH SPACE

Given a finite-dimensional Banach space and an Auerbach basis of , it is proved that there exist linear combinations of with coordinates , such that , for , and , for .

Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems

Abstract In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

Characteristic properties of the Gurariy space

The Gurariy space G is defined by the property that for every pair of finite dimensional Banach spaces L ⊂ M, every isometry T: L → G admits an extension to an isomorphism \(\mathop T\limits^ \sim :M \to G\) with ‖T‖‖T−1‖ ≤ 1 + ∈. We investigate the question when we can take \(\mathop T\limits^ \sim \) to be also an isometry (i.e., ∈ = 0). We identify a natural class of pairs L ⊂ M such that the above property for this class with ∈ = 0 characterises the Gurariy space among all separable Banach spaces. We also show that the Gurariy space G is the only Lindenstrauss space such that its finite-dimensional smooth subspaces are dense in all subspaces.

Reduced Convex Bodies in Finite Dimensional Normed Spaces: A Survey

For a convex body C in a finite dimensional real Banach space M d denote by $${\triangle(C)}$$ its thickness, i.e., its minimal width with respect to the norm. A convex body $${R \subset M^d}$$ is said to be reduced if $${\triangle(C) < \triangle(R)}$$ for each convex body C properly contained in R. The concept of reduced bodies is particularly useful for solving volume-minimizing problems in the area of convexity, and it is also important as extension of basic notions from convexity and functional analysis. Namely, on the one hand the class of reduced bodies is a “dualization” of the concept of complete sets (and, in Euclidean space, that of constant width). On the other hand, it forms a proper superset of the class of complete sets. We present the recent knowledge on this class of convex bodies in finite dimensional real Banach spaces. First we collect general properties of arbitrary reduced bodies. For example, we present constructions supporting our conjecture that in any normed space of dimension larger than 2 there are reduced bodies of unit thickness and diameter at least λ, for every positive number λ. Then we will lay special emphasize on reduced polytopes, and finally on the geometric description of planar reduced bodies. The survey also presents several research problems.

Incoherent systems and coverings in finite dimensional Banach spaces

We discuss the construction of coverings of the unit ball of a finite dimensional Banach space. There is a well-known technique based on comparing volumes which gives upper and lower bounds on covering numbers. However, this technique does not provide a method for constructing good coverings. Here we study incoherent systems and apply them to construct good coverings. We use the following strategy. First, we build a good covering using balls with a radius close to one. Second, we iterate this construction to obtain a good covering for any radius. We shall concentrate mainly on the first step of this strategy. Bibliography: 14 titles.

Some basic theorems on flat G chains

For a complete normed Abelian group G, we show that a flat G chain (respectively, normal flat G chain) of codimension 0 in a finite dimensional Banach space corresponds to a G valued functions that is Hm summable (respectively, of bounded variation). We also prove a constancy theorem for an m dimensional flat G chain T in a metric space assuming suppT is contained in the closure of a connected m dimensional Lipschitz submanifold M with M∩supp∂T=∅ and Hm(M)=Hm(ClosM)<∞.

On nonatomicity for non-additive functions

The notion of nonatomicity for set functions plays a key role in classical measure theory and its applications. For classical measures taking values in finite dimensional Banach spaces, it guarantees the connectedness of range. Even just replacing σ-additivity with finite additivity for measures requires some stronger nonatomicity property for the same conclusion to hold. In the present paper, we deal with non-additive functions – called ‘s-outer’ and ‘quasi-triangular’ – defined on rings and taking values in Hausdorff topological spaces. No algebraic structure is required on their target spaces. In this context, we make use of a notion of strong nonatomicity involving just the behavior of functions on ultrafilters of their underlying Boolean domains. This notion is proved to be equivalent to that proposed in earlier contributions concerning Lyapunov-types theorems in additive and non-additive frameworks. Thus, in particular, our analysis allows to generalize, improve and unify several known results on this topic.

Directional Lower Porosity

We investigate differences between upper and lower poros- ity. In finite dimensional Banach spaces every upper porous set is direc- tionally upper porous. We show the situation is very different for lower porous sets; there exists a lower porous set in R 2 which is not even a countable union of directionally lower porous sets.

Another Proof for Non-Supercyclicity in Finite Dimensional Complex Banach Spaces

We give an elementary proof, based on linear algebra and on a simple and well-known technique from the theory of dynamical systems, for the non-existence of supercyclic linear operators defined on a finite dimensional complex Banach space with dimension greater than or equal to two.

Bounded representation and radial projections of bisectors in normed spaces

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion com- bines the concepts of bisector and shadow boundary of the unit ball, both corresponding with the same spatial direction. The bounded representation visualizes the connection be- tween the topology of bisectors and shadow boundaries (Lemma 1) and gives the possibility to simplify and to extend some known results on radial projections of bisectors. Our main result (Theorem 1) says that in the manifold case the topology of the closed bisector and the topology of its bounded representation are the same; they are closed, (n − 1)-dimensional balls embedded in Euclidean n-space in the standard way.

On universal covers in normed planes

Abstract In 1914, Lebesgue posed his famous universal cover problem for the Euclidean plane which is still unsolved. We present the first explicit investigation of this problem for arbitrary normed planes. A convex body K in a finite dimensional real Banach space is called universal cover if any set of diameter 1 can be covered by a congruent copy of K; if ”congruent copy“ is replaced by ”translate”, then K is called strong universal cover. We describe the smallest regular hexagons, the smallest equilateral triangles, and the smallest squares (all these figures defined in the sense of the norm) that are strong universal covers in normed planes. In addition, the paper contains a new characterization of Radon planes. In view of universal covers having ball shape we shortly discuss also Jung constants of real Banach spaces, and a known result on Borsuk’s partition problem is reproved.

The log-Brunn–Minkowski inequality

For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces

We prove that an onto isometry between unit spheres of finite-dimensional polyhedral Banach spaces extends to a linear isometry of the corresponding spaces.

Minsum hyperspheres in normed spaces

We study the minsum hypersphere problem in finite dimensional real Banach spaces: given a finite set D of (positively weighted) points in an n-dimensional normed space (n≥2), find a minsum hypersphere, i.e., a homothet of the unit sphere of this space that minimizes the sum of (weighted) distances between the hypersphere and the points of D.We show existence results of the following type: there are situations where minsum hyperspheres do not exist, no point-shaped hypersphere can be optimal, and for any norm there exists a set of points D such that a hyperplane is better than any proper hypersphere. We also prove that the intersection of a minsum hypersphere S and conv(D) is non-empty, that D⊆conv(S) implies |S∩conv(D)|≥2, and that |S∩conv(D)|<∞ implies S∩conv(D)⊆D. A certain halving criterion regarding the sums of weights inside and outside of S is verified, and various further results are obtained for large classes of norms, like strictly convex, smooth, and polyhedral norms.

The logarithmic Minkowski problem

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

Minimal enclosing discs, circumcircles, and circumcenters in normed planes (Part II)

Until now there are almost no results on the precise geometric location of minimal enclosing balls of simplices in finite-dimensional real Banach spaces. We give a complete solution of the two-dimensional version of this problem, namely to locate minimal enclosing discs of triangles in arbitrary normed planes. It turns out that this solution is based on the classification of all possible shapes that the intersection of two norm circles can have, and on a new classification of triangles in normed planes via their angles. We also mention that our results are closely related to basic notions like coresets, Jung constants, the monotonicity lemma, and d-segments.