The Local Steiner Problem in Finite-Dimensional Normed Spaces

We develop a general method for proving that certain star configurations in finite-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlor and Morgan (1994) that could only be applied to differentiable norms. The generalization uses the subdifferential calculus from convex analysis. We apply this method to two special norms. The first norm, occurring in the work of Cieslik, has unit ball the polar of the difference body of the n-simplex (in dimension 3 this is the rhombic dodecahedron). We determine the maximum degree of a given point in a Steiner minimal tree in this norm. The proof makes essential use of extremal finite set theory. The second norm, occurring in the work of Conger (1989), is the sum of the ℓ₁-norm and a small multiple of the ℓ₂ norm. For the second norm we determine the maximum degree of a Steiner point.     

$\beta$-齐性赋范空间中近似二次泛函不等式

Using the direct method,we investigate the generalized Hyers-Ulam stability of the following quadratic functional inequality■inβ-homogeneous complex Banach spaces.     

Isometries on Noncommutative Symmetric Spaces

Doklady Mathematics - Let $$\mathcal{M}$$ be an atomless semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ (or else, an atomic von Neumann algebra with all...     

Finite-Dimensional Bicomplex Hilbert Spaces

This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including the spectral decomposition theorem. Applications to concepts relevant to quantum mechanics, like the evolution operator, are pointed out.     

AM-Totally Boundedness on Normed Riesz Spaces

In this paper we introduce and study the class of AM-totally bounded operators. Also we introduce the notion of graph lattice norm to prove a domination result for this class of operators. In particular, we extend the main theorem of Aqzzouz et al. [8]. Further, we give new results concerning the power domination problem by a totally bounded operator.

     

Linear Isometries of S~p Spaces

Abstract This paper gencralizes the result about linear isometries of S~ spaces given by W.P.Novingerand D.M.Oberlin[2]for the unite dise of C to the bounded symmetric domains of C~n     

Tiling Infinite-Dimensional Normed Spaces

The present paper deals with real infinite-dimensional normedspaces; some of the main concepts here make sense, and havebeen treated in the literature, in the general context of topologicalHausdorff linear spaces over reals. A subset of a normed space X is a body if it is different fromX itself and is the closure of its non-empty interior. A coveringof X by bodies is called a tiling of X whenever any two differentmembers of it have disjoint interiors. The elements of sucha covering are called tiles. A tiling is bounded (respectivelyconvex) whenever each tile is bounded (respectively convex).1991 Mathematics Subject Classification 46B20.     

Linear Isometries of S^D Spaces

This paper generalizes the result about linear isometries of S^pspaces given by W.P.Novinger and D.M.Oberlin[2] for the unite dise of C to the bounded symmetrie domains of C^n。     

Affine Variational Inequalities on Normed Spaces

This paper studies infinite-dimensional affine variational inequalities on normed spaces. It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities. We present two basic facts about infinite-dimensional affine variational inequalities: the Lagrange multiplier rule and the solution set decomposition.     

Mazur Sets in Normed Spaces

In this paper we investigate different questions concerning Mazur sets in normed spaces, which point out the close connections between geometric functional analysis and discrete geometry. Motivated by a result of Chen and Lin, we study the relationship between Mazur disks and weak* denting points of the dual unit ball. We prove that the only Mazur sets of the spaces l₁ⁿ are points and closed balls. Finally, a new stability property for the family of all sets which are intersections of closed balls is found.     

Subspaces of Normed Riesz Spaces

It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm p satisfies p(x)p(y) for all x and y with -yxy. A similar characterization of the subspaces of M-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the r-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given.     

序列空间上的可乘等距与渐进可乘等距

本文证明了（l^βn）的单位球面间的非满等距在一定条件下可被线性等距延拓至全空间。并刻画了l^p（P〉0）上的非满渐进可乘等距算子．     

Extremal Homogeneous Polynomials on Real Normed Spaces

IfPis a continuousm-homogeneous polynomial on a real normed space andPis the associated symmetricm-linear form, the ratio P/P always lies between 1 andm^m/m!. We show that, as in the complex case investigated by Sarantopoulos (1987,Proc. Amer. Math. Soc.99, 340–346), there areP's for which P/P=m^m/m! and for whichPachieves norm if and only if the normed space contains an isometric copy of ℓ^m₁. However, unlike the complex case, we find a plentiful supply of such polynomials providedm4.     

Sobolev Spaces with only Trivial Isometries

We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries.     

On Bisectors in Minkowski Normed Spaces

We discuss the concept of the bisector of a segment in a Minkowski normed n-space, and prove that if the unit ball K of the space is strictly convex then all bisectors are topological images of a hyperplane of the embedding Euclidean n-space. The converse statement is not true. We give an example in the three-space showing that all bisectors are topological planes, however K contains segments on its boundary. Strict convexity ensures the normality of Dirichlet-Voronoi-type K-subdivision of any point lattice.     

Spaces Universal under Closed Embeddings for Finite-Dimensional Complete Metric Spaces

We shall prove that for every natural number n and every cardinalnumber there exists an n-dimensional complete metric spaceX_n, of weight such that every n-dimensional complete metricspace of weight is embeddable in X_n, as a closed subset.     

Differential Vector Variational Inequalities in Finite-Dimensional Spaces

In this paper, a differential vector variational inequality is introduced and studied in finite-dimensional Euclidean spaces. The existence of a Carathéodory weak solution for the differential vector variational inequality is presented under some suitable conditions. Furthermore, the upper semicontinuity and the lower semicontinuity of the solution sets for the differential variational inequality are established when both the mapping and the constraint set are perturbed by two different parameters.     

Isometries of Weighted Spaces of Harmonic Functions

We investigate the isometries between weighted spaces of harmonic functions. We show that, under some mild conditions, every isometry is a composition operator. Our research shows that the structure of isometries of weighted spaces of harmonic functions is, in general, simpler than that observed for weighted spaces of holomorphic functions. Supported by MEC and FEDER Project MTM 2005-08210.