Characterizing Starshaped Sets by Maximal Visibility

Let S be a nonempty set in a real topological linear space L. p S is a point of maximal visibility of S if and only if it admits a neighbourhood N in L such that Sq Sp for every point q S N, where Sx = { s S : x is visible from s via S }. For S being either open and connected or the closure of its connected interior, it is shown that the kernel of S is the set of all maximal visibility points of S. Planar examples reveal that the topological assumptions on S are necessary. This substantially strengthens a recent result of Toranzos and Forte Cunto.     

Some results on best approximation in spaces with convex structure

Some generalizations of the result proved by S.P Singh[J.Approx.Theory 25(1979),89-90]are presented in convex metric spaces.The results proved contain several known results on the subject.     

SOME RESULTS ON BEST APPROXIMATION IN SPACES WITH CONVEX STRUCTURE

Some generalizations of the result proved by S.P. Singh [J. Approx. Theory 25（1979）, 89-90] are presented in convex metric spaces. The results proved contain several known results on the subject.     

Some Characterization of Locally Nonconical Convex Sets

A closed convex set Q in a local convex topological Hausdorff spaces X is called locally nonconical (LNC) if for every x, y Q there exists an open neighbourhood U of x such that . A set Q is local cylindric (LC) if for x, y Q, x y, z (x, y) there exists an open neighbourhood U of z such that U Q (equivalently: bd(Q) U) is a union of open segments parallel to [x, y]. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication LNC LC was proved in general, while the inverse implication was proved in case of Hilbert spaces.     

Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

We study solutions of the problem (0.1) where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if D₀ and D₁ are starshaped with respect to the same point , then the same occurs for every superlevel set of u.     

On a Theorem of Deutsch and Singer

We introduce the notion of a starshaped set-valued function, and relate it to convex set-valued functions. We prove a theorem which, in the convex case, implies that single-valuedness is exceptional if the function is not everywhere single-valued. A second result shows a remarkable costantness of images for starshaped set-valued functions. Both theorems were inspired from a result of Deutsch and Singer, which they strengthen.     

Partial Orders on Weak Orders Convex Subsets

We study a visibility relation on the nonempty connected convex subsets of a finite partially ordered set and we investigate the partial orders representable as a visibility relation of such subsets of a weak order. Moreover, we consider restrictions where the subsets of the weak order are total orders or isomorphic total orders.     

An Optimal Krasnosel'skii-type Theorem for an Open Starshaped Set

It is shown that for an open connected nonconvex set S in a real topological linear space ker conv , where slnc S denotes the set of strong local nonconvexity points of S and % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa % aaleaacaWG6baabeaakiabg2da9iaacUhacaWGZbGaeyicI4Saam4u % aiaacQdacaWG6baaaa!3EFE! \[ A_z = \{ s \in S:z \] is clearly visible from s viaS . This formula, combined with familiar procedures, generates the Krasnosel'skii-type characterizations for the dimension of the kernel of S in complete separable real metric linear spaces and, provided slncS is bounded, in finite-dimensional Euclidean spaces. The latter generalizes a planar result and settles an open problem.     

弱紧局部一致凸空间中的度量投影

本文研究了弱紧局部一致凸空间中度量投影的弱连续性,将Ｂ．Ｂ．Ｐａｎｄａ和Ｏ．Ｐ．Ｋａｐｏｏｒ的相应结论推广到更一般的弱紧局部一致凸空间．最后给出了局部一致凸点的一个必要条件．     

Structure of closed finitely starshaped sets

A set is finitely starshaped if any finite subset of is totally visible from some point of . It is well known that in a finite-dimensional linear space, a closed finitely starshaped set which is not starshaped must be unbounded. It is proved here that such a set must admit at least one direction of recession. This fact clarifies the structure of such sets and allows the study of properties of their visibility elements, well known in the case of starshaped sets. A characterization of planar finitely starshaped sets by means of its convex components is obtained. Some plausible conjectures are disproved by means of counterexamples.     

On the Geometry of Locally Nonconical Convex Sets

A convex subset Q of a Hausdorff topological vector space is called locally nonconical (LNC) if for every two points x,yQ there is a relative neighborhood U of x in Q such that U+ (y-x) Q. A geometric characterization (Theorem 2.2) of closed LNC sets with nonempty interior in a Hilbert space is supplied. It states that any proper line segment ]x,y[ contained in bd(Q), the topological boundary of Q, lies inside a relative neighborhood in bd(Q) composed of parallel line segments. It is shown that one half of this characterization, at least, generalizes to the setting of a locally convex Hausdorff topological vector space (LCHTVS). This leads to the observation that the set ext(Q) of extreme points of any LNC set Q in an LCHTVS is closed. Finally, it is proven that, in the same setting, all LNC sets are uniformly stable and, hence, stable.     

Characterization of balls by generalized Riesz energy

We show that balls, circles and 2‐spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution.     

On a discrete version of alexandrov’s projection theorem

In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.     

非奇H矩阵的等价条件

1引言H矩阵与M矩阵是计算数学中应用很广的一类矩阵类,在数理方程、控制论、电力系统、经济数学等众多领域中都有广泛的应用。然而对它的判别非常困难,所以对该类矩阵的判别具有非常重要的意义．王川龙,李继成,王广彬,黄廷祝等在[1-6]中分别给出了一系列实用性的判别准则,但都是充分条件,本文给出了非奇H矩阵的充要条件．     

Banach空间上凸集的凸性

以Banach空间的一般凸集为研究对象,将Banach空间的凸性研究推广到了内部非空的凸集上．打破了从单位球出发研究Banach空间几何的具有局限性的研究方法,给出了严格凸集的若干特征刻画及性质,并得到了严格凸集和光滑集之间的对偶定理．     

New Combinations of Convex Sets

The family of convex sets in a (finite dimensional) real vector space admits several unary and binary operations – dilatation, intersection, convex hull, vector sum – which preserve convexity. These generalize to convex functions, where there are in fact further operations of this kind. Some of the latter may be regarded as combinations of two such operations, acting on complementary subspaces. In this paper, a general theory of such mixed operations is introduced, and some of its consequences developed.     

Local properties of inexact methods for minimizing nonsmooth composite functions

Methods for minimization of composite functions with a nondifferentiable polyhedral convex part are considered. This class includes problems involving minimax functions and norms. Local convergence results are given for “active set” methods, in which an equality-constrained quadratic programming subproblem is solved at each iteration. The active set consists of components of the polyhedral convex function which are active or near-active at the current iteration. The effects of solving the subproblem inexactly at each iteration are discussed; rate-of-convergence results which depend on the degree of inexactness are given.     

凸集上投影算子的一个单调性质

本文证明了投影算子的一个新的单调性质，并简要讨论了与其它单调性质的关系     

On the Subdifferentiability of the Difference of Two Functions and Local Minimization

We set up a formula for the Fréchet and ε-Fréchet subdifferentials of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. As a consequence of this formula, we give necessary and sufficient conditions for local optimality in nonconvex optimization. Our analysis relies on the notion of gap continuity of multivalued maps and involves concepts of independent interest such as the notions of blunt and sharp minimizers and the notion of equi-subdifferentiability.