Closedness of a Convex Cone and Application by Means of the End Set of a Convex Set

This article presents new conditions that ensure the closedness of a convex cone in terms of the end set and the extent of its generator. The results significantly extend the classic condition. The new closedness conditions are utilized to obtain a simple formula of the least global error bound and a suitable regularity condition of the set containment problem for sublinear functions.     

Directional End of a Convex Set: Theory and Applications

A point of a convex set belongs to its end in a given direction when this direction is not feasible at that point. This paper analyzes the properties of the directional end of general convex sets and closed convex sets (for which the directional ends are connected by arcs) as well as the relationship between the directional end and certain concepts on the illumination of convex bodies. The paper includes applications of the directional end to the theory of linear systems.     

Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications

In this paper, we characterize the nonemptiness and compactness of the set of weakly efficient solutions of a convex vector optimization problem with cone constraints in terms of the level-boundedness of the component functions of the objective on the perturbed sets of the original constraint set. This characterization is then applied to carry out the asymptotic analysis of a class of penalization methods. More specifically, under the assumption of nonemptiness and compactness of the weakly efficient solution set, we prove the existence of a path of weakly efficient solutions to the penalty problem and its convergence to a weakly efficient solution of the original problem. Furthermore, for any efficient point of the original problem, there exists a path of efficient solutions to the penalty problem whose function values (with respect to the objective function of the original problem) converge to this efficient point.     

Packings of Unequal Circles in a Convex Set

   Abstract. In the Euclidean plane let T be a convex set, and let K ₁ , ..., K _n be a family of n ≥ 2 circles packed into T . We show that the density of each such packing is smaller than

锥凸集值映射的基本性质

本文首先在R~m的幂集上定义了一种锥序关系并借助这种序关系定义锥凸集值映射,证明了普通单值凸函数的一些基本性质拓广到这种锥凸集值映射时仍成立.     

On Criteria for Closedness of an Attainable Set of a Discrete Neutral Control System

This paper deals with the closedness of the attainable set in the function space W⁽¹⁾₂ × L₂ of a linear neutral system with commensurate delays in state and control. Necessary and sufficient algebraic criteria expressed in terms of the system matrices for the closedness are derived. By the additional restriction that the control is a scalar control, another criterion for closedness, based on the transfer matrix, is indicated and used to verify the results.     

锥有效拟凹集的有效点集的连通性

本文引进了锥有效拟凹集的概念，讨论了R ^m-严格拟凹与R ^m－有效拟凹的关系，证明了一个紧集为锥有效拟凹且其有效点集为闭集时，这个有效点集是连通的。     

模糊Choquet-可积函数空间的凸锥结构

在一般有限模糊测度空间上研究了非负模糊Choquet-可积函数的模糊测度表示, 进而获得了非负模糊Choquet-可积函数空间和一般模糊Choquet-可积函数空间构成凸锥的充要条件.     

The Determination of a Convex Set from Its Angle Function

In this paper we discuss the followingquestion: how can we decide whether a convex setis determined by its angle function or not? We givesufficient conditions for convex polygons and forregular convex sets which guarantee that the setis distinguishable. We also investigate the question: which setsare typical (in the sense of Baire category), thosewhich are distinguishable or those which are not?We prove that the family of distinguishable sets is ofsecond Baire category.     

Sharpening an Estimate of the Size of the Sumset of a Convex Set

A finite set A = {a₁ < … <a_n}? ? is said to be convex if the sequence (a_i ? a_i?1)ⁿ_i=2 is strictly increasing. Using an estimate of the additive energy of convex sets, one can estimate the size of the sumset as ∣A + A∣ ? ∣A∣^102/65, which slightly sharpens Shkredov’s latest result ∣A + A∣ ? ∣A∣^58/37.

     

Minimum Distance to the Complement of a Convex Set: Duality Result

The subject of this paper is to study the problem of the minimum distance to the complement of a convex set. Nirenberg has stated a duality theorem treating the minimum norm problem for a convex set. We state a duality result which presents some analogy with the Nirenberg theorem, and we apply this result to polyhedral convex sets. First, we assume that the polyhedral set is expressed as the intersection of some finite collection of m given half-spaces. We show that a global solution is determined by solving m convex programs. If the polyhedral set is expressed as the convex hull of a given finite set of extreme points, we show that a global minimum for a polyhedral norm is obtained by solving a finite number of linear programs.     

几种凸模糊集间的转换关系

在引用扎德所定义的凸模糊集、强凸模糊集、严格凸模糊集等概念的基础上,探讨了这三种凸模糊集间的转换条件,得到凸模糊集与强凸模糊集、强凸模糊集与严格凸模糊集间的等价条件。     

Transversals for Families of Translates of a Two-Dimensional Convex Compact Set

The following Theorem 1 gives an affirmative answer to Grünbaum's old question. Let F be a family of translates of a convex compact set K⊂R ² . If every two elements of F have a common point, then there exist three points A, B, C ∈ R ² such that every element of F contains some of these points. Received December 28, 1998, and in revised form October 20, 1999. Online publication May 15, 2000.     

A Note on Convex Decompositions of a Set of Points in the Plane

For any set P of n points in general position in the plane there is a convex decomposition of P with at most elements. Moreover, any minimal convex decomposition of such a set P has at most elements, where k is the number of points in the boundary of the convex hull of P.Partially supported by Conacyt, MexicoFinal version received: November 10, 2003     

Estimates for Integral Means of Hyperbolically Convex Functions

We prove the Mejia-Pommerenke conjecture that the Taylor coefficients of hyperbolically convex functions in the disk behave like O(log^?2(n)/n) as n → ∞ assuming that the image of the unit disk under such functions is a domain of bounded boundary rotation. Moreover, we obtain some asymptotically sharp estimates for the integral means of the derivatives of such functions and consider an example of a hyperbolically convex function that maps the unit disk onto a domain of infinite boundary rotation.     

关于平面19-点集的空凸分划问题

主要研究了平面上处于一般位置的19-点集,根据其凸包边数的不同,分别讨论了其所含空凸多边形的个数,得出G(19)≤5.在此基础上,对平面上处于一般位置的n-点集得出G(n)≤[11n/42],从而改进了G(n)的上界.