非常极凸空间的推广及其对偶概念

本文研究了k非常极凸和k非常极光滑空间的问题.利用Banach空间理论的方法,证明了k非常极凸空间和k非常极光滑空间是一对对偶概念,并且k非常极凸空间(k非常极光滑空间)是严格介于k一致极凸空间和k非常凸空间(k一致极光滑空间和k非常光滑空间)之间的一类新的Banach空间,得到了k非常极凸空间和k非常极光滑空间的若干等价刻画以及k非常极凸(k非常极光滑性)与其它凸性(光滑性)之间的蕴涵关系,推广了非常极凸空间和非常极光滑空间,完善了k非常极凸空间及其对偶空间的研究.     

非常极凸空间

本文使用非常极凸的定义，证明了非常极凸和非常光滑是互为对偶空间且严格介于弱k凸和非常凸之间的空间，最后得到了非常极凸的一些特征．     

fc-非常极凸空间的一些特征

本文研究了k-非常极凸空间的问题,利用k维体积定义了k-非常极凸空间,使用k-非常极凸的概念,得到了k-非常极凸空间的性质和一些特征,推广了k-drop凸空间.     

关于k-强极凸空间

本文证明k-强极凸是严格介于k-非常极凸和k-极凸之间的凸性.利用k-强极凸的概念,得到k-强极凸的一些特征.     

局部凸空间的k-一致极凸性与k-一致极光滑性

首先引入局部凸空间的k-一致极凸性和k-一致极光滑性这一对对偶概念,它们既是Banach空间k-一致极凸性和k-一致极光滑性推广,又是局部凸空间一致极凸性和一致极光滑性的自然推广.其次讨论它们与其它k-凸性(k-光滑性)之间的关系.最后,在P-自反的条件下给出它们之间的等价对偶定理.     

Orlicz-Sobolev空间的k一致凸性

k一致凸性是Banach空间的重要几何属性,结合Orlicz空间和Sobolev空间的技巧给出了Orlicz-Sobolev空间关于Luxemburg范数的k一致凸性成立的充要条件.     

关于Banach空间k一致凸及k一致光滑性

用统一且简洁形式刻画、定义了Banach空间的(局部)k一致凸、k-强凸、ω-强凸性.给出(局部)k一致光滑性概念,并讨论了上述空间的关系及性质.     

关于凸体的Lp-曲率映象的不等式

Lutwak提出了凸体的Lp-曲率映象的概念,并证明了凸体与其Lp-曲率映象的体积之间的一个不等式.本文给出了Lutwak结果的一个一般形式,继而证明了凸体与其Lp-曲率映象的极的体积之间的一个不等式,并得到了凸体的Lp-投影体和Lp-曲率映象的体积之间的一个不等式.     

凸体间几种仿射不变距离的等价性与估计

本文研究了凸体间(绝对)Banach-Mazur距离各种不同的定义,证明了它们的等价性;给出了Banach-Mazur距离与绝对Banach-Mazur距离相等的一个充分条件;最后研究了凸体极体间的BanachMazur距离,并对特殊凸体对证明了其Banach-Mazur距离与其某一对极体间的Banach-Mazur距离相等.文中结果为Banach-Mazur距离最佳上界的估计提供了进一步研究的基础.     

局部凸空间P-自反下的中点局部k-一致凸性(光滑性)

在局部凸空间已有的中点局部kk-一致凸性和中点局部k-一致光滑性这一对对偶概念的基础上,证明了中点局部kk-一致凸性与中点局部(k+1)-一致凸性的关系,给出了在P-自反的条件下它们之间的等价对偶定理.     

Fundamental results for pointfree convex geometry

Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: (1) the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of a point is more useful than the latter one from a category theoretic point of view and that the former notion of a point actually represents a polytope (or generic point) and the latter notion of a point properly represents a point. We also remark on the close relationships between pointfree convex geometry and domain theory.     

A note on integral-convexity in Banach spaces

In the present paper we focus on a generalization of the notion of integral convexity. This concept, introduced in [J.Y. Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J. Math. Anal. Appl. 295 (2004) 211-224] by replacing, in the definition of classical notion of convexity, the sum by the integral, has interesting applications in optimal control problems. By using, instead of Bochner integral, a more general vector integral, that of Pettis, we obtain some results on integral-extreme points of subsets of a Banach space stronger than those given in [J.Y. Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J. Math. Anal. Appl. 295 (2004) 211-224]. Finally, a natural example coming from measure theory is included, in order to reflect the relationships between different kinds of integral convexity.     

Convexities of metric spaces

We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.      

Conjugate Duality for Constrained Vector Optimization in Abstract Convex Frame

This article focuses on a conjugate duality for a constrained vector optimization in the framework of abstract convexity. With the aid of the extension for the notion of infimum to the vector space, a set-valued topical function and the corresponding conjugate map, subdifferentials are presented. Following this, a conjugate dual problem is proposed via this conjugate map. Then, inspired by some ideas in the image space analysis, some equivalent characterizations of the zero duality gap are established by virtue of the subdifferentials.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

The Strong Continuity of Convex Functions

A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.     

One class of separable optimization problems: solution method,application

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Craven's notion of K-invexity function (when K is a cone in ? ⁿ ) and Martin's notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.     

Strongly extreme points in Banach spaces

In this paper some properties of a special type of boundary point of convex sets in Banach spaces are studied. Specifically, a strongly extreme point x of a convex set S is a point of S such that for each real number r>0, segments of length 2r and centered x are not uniformly closer to S than some positive number d(x,r). Results are obtained comparing the notion of strongly extreme point to other known types of special boundary points of convex sets. Using the notion of strongly extreme point, a convexity condition is defined on the norm of the space under consideration, and this convexity condition makes possible a unified treatment of some previously studied convexity conditions. In addition, a sufficient condition is given on the norm of a separable conjugate space for every extreme point of the unit ball to be strongly extreme.     

K-drop凸空间中的性质

为了阐明何为K 强光滑空间的对偶空间 ,本文定义了K drop凸空间并且讨论了该空间的一些性质。同时借助K 强光滑空间的一个等价定义 ,证明了K drop凸空间与K 强光滑空间是对偶空间。文章最后用单位圆的切片给出了K drop凸空间的一等价命题 ,进而建立了K drop凸空间与drop性之间的关系。