关于弱拟凸集与广义凸集逼近的几点注记

本文证明了赋范线性空间中闭弱拟凸集必为凸集,并指出郭元明的”弱拟凸集的一些性质及其应用”、“广义凸集的联合逼近特性”两文中的主要结论实质上是已知的结果.     

平面常宽凸集的Firey-Sallee定理

常宽凸集是一类广泛应用在机械设计、医学等领域的特殊几何图形.本文探讨平面中的常宽凸集,简化证明著名的Firey-Sallee定理,即宽度相等的正Reuleaux多边形中Reuleaux三角形的面积最小.     

有界集空间中的几乎Chebyshev子集

研究了有界集关于一般集合的限制Chebyshev中心的存在唯一性。在集合的Hausdorff距离下,引进了有界集空间中的几乎Chebyshev子集的概念。证明了一致凸(自反局部一致凸)Banach空间中的任何闭子集都是关于有界集(紧凸子集)的几乎Chebyshev子集。     

伪凸集值映射的误差界

考虑了伪凸集值映射的误差界.证明了对于伪凸集值映射,局部误差界成立意味着整体误差界成立.通过相依导数,给出了伪凸集值映射存在误差界的一些等价叙述.     

近似凸集的某些性质

主要研究了两类近似凸集的关系和性质.首先,举例说明两类近似凸集没有相互包含关系.其次,在近似凸集(nearly convex)条件下,证明了在一定条件下函数上图是近似凸集与凸集的等价关系.同时,考虑了近似凸函数与函数上图是近似凸集的等价刻画、近似凸函数与函数水平集是近似凸集的必要性,并用例子说明近似凸函数与函数水平集是...     

平面上一类自相似集的Hausdorff测度与上凸密度

考虑平面单位正方形内生成的一类自相似集的Hausdorff测度的计算问题.在满足强分离条件及维数小于1的条件下,当相似比满足某些条件时,证明了自然覆盖为其实现上凸密度1计算的最好形状,因而自然覆盖即是最好覆盖.而作为它的直接推论得到该类自相似集的Hausdorff测度的精确值为(2s)~/(1/2),其中s为其Hausdorff维数.     

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.     

一致凸的线性度量空间（Ⅰ）

本文首先证明了一致凸的线性度量空间中的每个有界闻凸集都存在唯一的最佳逼近元，然后证明了一致凸的线性度量空间具有Ｈ性质。     

凸模糊子集的再定义

给出基于t-范上的凸模糊子集的定义,讨论这种凸模糊子集的同态像性质,重点研究了Tm-凸模糊子集的生成线,证明每一个TA－凸模糊子集为凸子集生成的Tm-凸模糊子集。     

灰色凸集及其基本性质

本文给出了灰色凸集的定义．并证明了灰色凸集具有一般凸集完全类似的性质．     

Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Polyominoes Determined by Permutations: Enumeration via Bijections

A permutominide is a set of cells in the plane satisfying special connectivity constraints and uniquely defined by a pair of permutations. It naturally generalizes the concept of permutomino, recently investigated by several authors and from different points of view [1, 2, 4, 6, 7]. In this paper, using bijective methods, we determine the enumeration of various classes of convex permutominides, including, parallelogram, directed convex, convex, and row convex permutominides. As a corollary we have a bijective proof for the number of convex permutominoes, which was still an open problem.     

Keplerian Action,Convexity Optimization,and the 4-Body Problem

In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints. Our approach is based on some novel features of the Keplerian action functional, constraint convex optimization techniques, and variational methods. We demonstrate the strength of this method by constructing relative periodic solutions for the planar four-body problem within a special topological class, and our results hold for an open set of masses.     

Total variation and Cheeger sets in Gauss space

The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context.     

Connectedness of the Efficient Set for Three-Objective Quasiconcave Maximization Problems

For three-objective maximization problems involving continuous, semistrictly quasiconcave functions over a compact convex set, it is shown that the set of efficient solutions is connected. With that, an open problem stated by Choo, Schaible, and Chew in 1985 is solved.     

Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems

In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem.     

Inner Approximation Method for a Reverse Convex Programming Problem

In this paper, we consider a reverse convex programming problem constrained by a convex set and a reverse convex set, which is defined by the complement of the interior of a compact convex set X. We propose an inner approximation method to solve the problem in the case where X is not necessarily a polytope. The algorithm utilizes an inner approximation of X by a sequence of polytopes to generate relaxed problems. It is shown that every accumulation point of the sequence of optimal solutions of the relaxed problems is an optimal solution of the original problem.     

An outcome space approach for generalized convex multiplicative programs

This paper addresses the problem of minimizing an arbitrary finite sum of products of two convex functions over a convex set. Nonconvex problems in this form constitute a class of generalized convex multiplicative problems. Convex analysis results allow to reformulate the problem as an indefinite quadratic problem with infinitely many linear constraints. Special properties of the quadratic problem combined with an adequate outer approximation procedure for handling its semi-infinite constrained set enable an efficient constraint enumeration global optimization algorithm for generalized convex multiplicative programs. Computational experiences illustrate the proposed approach.     

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.