Cassini Curves in Normed Planes

We extend the concept of Cassini curves from the Euclidean plane to normed (or Minkowski) planes and show that geometric properties of (Minkowskian) Cassini curves are closely related to geometric properties of the unit disc determining the underlying normed plane.     

The Fermat–Torricelli Problem in Normed Planes and Spaces

We investigate the Fermat–Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat–Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat–Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach.     

On Some Rational Triangles

We prove that every set of n ≥ 3 points in \mathbbR²{\mathbb{R}^2} can be slightly perturbed to a set of n points in \mathbbQ²{\mathbb{Q}^2} so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily close to a given triangle are also considered. Given a triangle ABC, we show that for each ε > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < ε and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that A^￠￠, B^￠￠, C^￠￠ ? \mathbbQ²{A^{\prime\prime}, B^{\prime\prime}, C^{\prime\prime} \in \mathbb{Q}^2} and |AA′′|, |BB′′|, |CC′′| < ε.     

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.     

On an Orthogonality Equation in Normed Spaces

The aim of this paper is to give an answer to the question, posed by Alsina, Sikorska, and Tomás, as to whether each norm derivative preserving mapping is necessarily linear.     

On Sequences of Nested Triangles

Summary For a given triangle, we consider several sequences of nested triangles obtained via iterative procedures. We are interested in the limiting behavior of these sequences. We briefly mention the relevant known results and prove that the triangle determined by the feet of the angle bisectors converges in shape towards an equilateral one. This solves a problem raised by Trimble~[5].     

On Quasiconformal Mappings Between Hyperbolic Triangles

Quasiconformal mappings between hyperbolic triangles are considered.We give an explicit estimate of the dilation of the quasiconformal mappings,which generalize...     

概率赋范空间上的线性算子

本文提出了概率赋范空间上各类线性算子的概念,并仔细研究了它们之间的关系。     

On Free Planes in Lattice Ball Packings

This note, by studying the relations between the length of theshortest lattice vectors and the covering minima of a lattice,proves that for every d-dimensional packing lattice of ballsone can find a four-dimensional plane, parallel to a latticeplane, such that the plane meets none of the balls of the packing,provided that the dimension d is large enough. Nevertheless,for certain ball packing lattices, the highest dimension ofsuch ‘free planes’ is far from d.     

On Normed Lattices and Their Banach Completions

It is known that the Banach completion Y = bX of a normed lattice X need not preserve the properties to be Dedekind complete or σ-Dedekind complete. In this paper it is proved that the countable interpolation property and the property to be sequentially order complete are preserved under the Banach completion. To prove this results we found some sufficient conditions (which are close to necessary ones) on X which secure for Y to have the countable interpolation property and (respectively) to be sequentially order complete. These conditions are obtained with the help of the newly developed techniques based on representations of normed lattices. It is well known that order continuity, and σ-order continuity of a norm are preserved under the Banach completion. Here necessary and sufficient conditions on X to secure these properties in Y are discussed. Mathematics Subject Classification 2000: 46B42, 46E15     

关于任意三角形区域上的Durrmeyer算子

本文将区间[0,1]上的Durrmeyer算子推广到平面上的任意三角形区域中去,并在空间C(T),C^k(T)(k≥1),L_p(T)以及Sobolev空间W_pr(T)中研究了它的收敛性及逼近度估计,这里T是平面上的三角形区域。     

On the Best Approximation in Non-Archimedean Normed Linear Spaces

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On Sublinear Analogs of Weak Topologies in Normed Cones

Mathematical Notes -