Finite Dimensional Real Banach Spaces Research Articles

Multiplicative maps of matrices over rings

Let R be an arbitrary associative ring (not necessarily with an identity). A sufficient condition is given to determine when a multiplicative isomorphism on the matrix ring is additive without the assumption of idempotents. We formulate a version for matrix rings over rings with identities, and then present two applications. One is a new characterization of the automorphisms of the matrix algebra over an arbitrary field by means of multiplicative preservers, and the other one is a new characterization of topologically isomorphisms of finite dimensional real Banach spaces with dimensions no less than 2 by the existence of multiplicative isomorphisms between their rings of all bounded linear operators.

Duality of gauges and symplectic forms in vector spaces

A gauge $$\gamma$$ in a vector space X is a distance function given by the Minkowski functional associated to a convex body K containing the origin in its interior. Thus, the outcoming concept of gauge spaces $$(X, \gamma )$$ extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of X is even, then the fixation of a symplectic form yields an identification between X and its dual space $$X^*$$ . The image of the polar body $$K^{\circ }\subseteq X^*$$ under this identification yields a (skew-)dual gauge on X. In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur–Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.

Differential geometry of spatial curves for gauges

We derive Frenet-type results and invariants of spatial curves immersed in 3-dimensional generalized Minkowski spaces, i.e., in linear spaces which satisfy all axioms of finite dimensional real Banach spaces except for the symmetry axiom. Further on, we characterize cylindrical helices and rectifying curves in such spaces, and the computation of invariants is discussed, too. Finally, we study how translations of unit spheres influence invariants of spatial curves.

Computation of maximal projection constants

The linear projection constant Π(E) of a finite-dimensional real Banach space E is the smallest number C∈[0,+∞) such that E is a C-absolute retract in the category of real Banach spaces with bounded linear maps. We denote by Πn the maximal linear projection constant amongst n-dimensional Banach spaces. In this article, we prove that Πn may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the maximal relative projection constants of codimension n converge to 1+Πn. Furthermore, using the classification of K4-free two-graphs, we give an alternative proof of Π2=43. We also show by means of elementary functional analysis that for each integer n⩾1 there exists a polyhedral n-dimensional Banach space Fn such that Π(Fn)=Πn.

Geometry of Simplices in Minkowski Spaces

There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization, and the analogous definitions for normed spaces represent a promising topic. An example is the geometry of simplices in non-Euclidean normed spaces. We present new generalizations of well known properties of Euclidean simplices. These results refer to analogues of circumcenters, Euler lines, and Feuerbach spheres of simplices in normed spaces. Using duality, we also get natural theorems on angular bisectors as well as in- and exspheres of (dual) simplices.

Dichotomy of Iterated Means for Nonlinear Operators

In this paper, we discuss a dichotomy of iterated means of nonlinear operators acting on a compact convex subset of a finite-dimensional real Banach space. As an application, we study the mean ergodicity of nonhomogeneous Markov chains.

Ball convex bodies in Minkowski spaces

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed space is called ball convex if it coincides with its ball hull, which is obtained as intersection of all balls (of fixed radius) containing $S$. Ball convex sets are closely related to notions like ball polytopes, complete sets, bodies of constant width, and spindle convexity. We will study geometric properties of ball convex bodies in normed spaces, for example deriving separation theorems, characterizations of strictly convex norms, and an application to complete sets. Our main results refer to minimal representations of ball convex bodies in terms of their ball exposed faces, to representations of ball hulls of sets via unions of ball hulls of finite subsets, and to ball convexity of increasing unions of ball convex bodies.

On symmetry of Birkhoff-James orthogonality of linear operators on finite-dimensional real Banach spaces

We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space $ \mathbb{X},$ which answers a question raised recently by one of the authors in \cite{S} [D. Sain, \textit{Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, Journal of Mathematical Analysis and Applications, accepted, $ 2016 $}]. We prove that $ T\in B(\mathbb{X}) $ is left symmetric if and only if $ T $ is the zero operator. If $ \mathbb{X} $ is two-dimensional then the same characterization can be obtained without the smoothness assumption. We also explore the properties of right symmetric linear operators defined on a finite dimensional real Banach space. In particular, we prove that smooth linear operators on a finite-dimensional strictly convex and smooth real Banach space can not be right symmetric.

Angles in normed spaces

The concepts of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to the Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including further related aspects, as well. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.

Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces

In this paper we characterize Birkhoff–James orthogonality of linear operators defined on a finite dimensional real Banach space X. We also explore the left symmetry of Birkhoff–James orthogonality of linear operators defined on X. Using some of the related results proved in this paper, we finally prove that T∈L(lp2) (p≥2,p≠∞) is left symmetric with respect to Birkhoff–James orthogonality if and only if T is the zero operator.

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.

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.

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.

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.

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.

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

It is surprising that there are almost no results on the precise location of (all) minimal enclosing balls, circumballs, and circumcenters of simplices in finite-dimensional real Banach spaces. In this paper and a subsequent second part of it we give the starting point in this direction, also for computational investigations. More precisely, we present the first thorough study of these topics for triangles in arbitrary normed planes. In the present Part I we lay special emphasize on a complete description of possible locations of the circumcenters, and as a needed tool we give also a modernized classification of all possible shapes of the intersection that two homothetic norm circles can create. Based on this, we give in Part II the complete solution of the strongly related subject to find all minimal enclosing discs of triangles in arbitrary normed planes.

The numerical radius of Lipschitz operators on Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces. We give its basic properties. Our main result is a characterization of finite-dimensional real Banach spaces with Lipschitz numerical index&nbsp;$1$. We also explicitly compute the Lip

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.

On regular 4-coverings and their application for lattice coverings in normed planes

It is well known that the famous covering problem of Hadwiger is completely solved only in the planar case, i.e.: any planar convex body can be covered by four smaller homothetical copies of itself. Lassak derived the smallest possible ratio of four such homothets (having equal size), using the notion of regular 4-covering. We will continue these investigations, mainly (but not only) referring to centrally symmetric convex plates. This allows to interpret and derive our results in terms of Minkowski geometry (i.e., the geometry of finite dimensional real Banach spaces). As a tool we also use the notion of quasi-perfect and perfect parallelograms of normed planes, which do not differ in the Euclidean plane. Further on, we will use Minkowskian bisectors, different orthogonality types, and further notions from the geometry of normed planes, and we will construct lattice coverings of such planes and study related Voronoi regions and gray areas. Discussing relations to the known bundle theorem, we also extend Miquel’s six-circles theorem from the Euclidean plane to all strictly convex normed planes.