Regularity of Sets with Quasiminimal Boundary Surfaces in Metric Spaces

This paper studies regularity of perimeter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincaré inequality. The main result shows that the measure-theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory to study rectifiability issues related to quasiminimal sets in the strong A _∞-weighted Euclidean case.     

Best Approximation on Convex Sets in Metric Linear Spaces

In this paper the concepts of strictly convex and uniformly convex normed linear spaces are extended to metric linear spaces. A relationship between strict convexity and uniform convexity is established. Some existence and uniqueness theorems on best approximation in metric linear spaces under different conditions are proved.     

Variational Convergence of Gradient Flows and Rate-Independent Evolutions in Metric Spaces

We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p >?1, when the exponents p converge to 1.     

Quasiopen Sets,Bounded Variation and Lower Semicontinuity in Metric Spaces

Potential Analysis - In the setting of a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we show that the total variation of functions of...     

Fine Properties of Sets of Finite Perimeter in Doubling Metric Measure Spaces

The aim of this paper is to study the properties of the perimeter measure in the quite general setting of metric measure spaces. In particular, defining the essential boundary ^* E of E as the set of points where neither the density of E nor the density of XE is 0, we show that the perimeter measure is concentrated on ^* E and is representable by an Hausdorff-type measure.     

Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N^1,1-spaces) and the theory of heat semigroups (a concept related to N^1,2-spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting.     

Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned. (Received 21 February 2000; in revised form 19 October 2000)     

Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned.     

度量空间和锥度量空间中的若干不动点定理

给出了度量空间和锥度量空间中的若干不动点定理.利用这些不动点定理,统一并推广了度量空间和锥度量空间中的若干经典的不动点定理.     

Banach空间集值映射的度量正则性与变分方程的Lipschitz稳定性

综述了集值映射的某些概念,例如度量正则性、伪Lipschitz性质(Aubin性质)、度量次正则性和Calm性质和这些概念的相互关系以及某些判据.也给出了他们在变分方程解的鲁棒Lipschitz稳定性、约束优化问题的最优性条件、集合族的线性正则性质和广义方程迭代过程的收敛性.     

概率度量空间中的Ekeland变分原理与集值映象的Caristi重合定理

借助偏序方法,本文得到概率度量空间中之一推广形式的Ekeland变分原理及一集值形式的Caristi重合定理,同时证明了这两个定理之间的等价性.本文结果是[1,2,5,6,7,9]中相应结果的改进和推广.     

Variational Principles, Minimization Theorems, and Fixed-Point Theorems on Generalized Metric Spaces

In this paper, we prove a new minimization theorem by using the generalized Ekeland variational principle. We apply our minimization theorem to obtain some fixed-point theorems. Our results extend, improve, and unify many known results due to Kui, Ekeland, Takahashi, Caristi, iri, and others.     

Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces

By employing the notion of exceptional family of elements, we establish some existence results for generalized variational inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability for the pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is strictly feasible.     

非自反Banach空间中的度量投影

该文给出非自反Banach空间中一类超平面上度量投影的表达式．在近严格凸Banach空间中,研究了它们的连续性．对于对偶Banach空间X*,给出弱*闭子集上度量投影的一些连续性结果．     

Sobolev Spaces with Zero Boundary Values on Metric Spaces

We generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure. We show that many classical results extend to the metric setting. These include completeness, lattice properties and removable sets.     

The Completeness of Cone Metric Spaces

Some topological properties of cone metric spaces are discussed and proved that every cone metric space （X, d） is complete if and only if every family of closed subsets of X which has the finite intersection property and which for every c ε E, 0 〈〈 c contains a set of diameter less that c has non-empty intersection.     

Spaces S p with Nonsymmetric Metric

We determine exact values of the best approximations and Kolmogorov widths of q-ellipsoids in spaces defined by anisotropic metric.     

Volume in General Metric Spaces

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be “faithfully preserved” by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of \(k\) points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set \(S = \{v_1, \ldots , v_k \}\) of points in a general metric space: the product of the distances from \(v_i\) to \(\{ v_1, \dots , v_{i-1} \}\) , normalized by \(\tfrac{1}{(k-1)!}\) , where the ordering of the points is that given by Prim’s minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of \(S\) into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige’s definition: the use of any other order in the product affects volume \(^{1/(k-1)}\) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.     

Best n-Term Approximations in Spaces with Nonsymmetric Metric

We determine exact values of n-term approximations of q-ellipsoids in the spaces .