锥拟凸集值映射与集值映射的锥半连续性

本文借助锥界集定义了一种新的锥拟凸集值映射,并且利用广义Minkowski泛函讨论了锥半连续及锥外上半连续集值映射的锥拟凸性．     

集值映射的一种锥凸性及标量化

在一种集合偏序关系下提出了集值映射的标量锥拟凸概念, 讨论了它与各种锥凸性的关系. 然后对恰当锥拟凸性得到了某种水平集意义下的刻画. 同时建立了集值映射的各种锥凸性通过实值单调增加凸函数表示的标量化复合法则. 最后给出了利用Gerstewitz泛函表示的对集值映射的锥拟凸性的标量化刻画.     

集值映射的一种锥凸性及标量化

在一种集合偏序关系下提出了集值映射的标量锥拟凸概念,讨论了它与各种锥凸性的关系.然后对恰当锥拟凸性得到了某种水平集意义下的刻画.同时建立了集值映射的各种锥凸性通过实值单调增加凸函数表示的标量化复合法则.最后给出了利用Gerstewitz泛函表示的对集值映射的锥拟凸性的标量化刻画.     

预不变凸模糊集的一些性质

在下半连续的条件下,给出了一个模糊集是预不变凸模糊集的充分条件,并将模糊凸集上相关性质在模糊不变凸集上作了相应的推广.所给出的条件和证明都是新的.     

有界闭凸值测度的集值积分

引入实值函数关于有界闭凸值测度的集值积分,并讨论了集值积分的收敛定理,证明了当集值测度为有界闭凸集值的有界变差集值测度时,关于弱紧凸集值测度的积分性质对有界闭凸集值测度仍然保持.推广了实值函数关于弱紧凸值测度的积分.     

具有β-中点性质的非β-凸集(0<β<1)

首先给出具有中点性质1/2x+1/2y∈A,x,y∈A的开集或者闭集均是凸集的完整证明,接着通过给出满足β-中点性质1/(2~(1/β))x+1/(2~(1/β))y∈A,x,y∈A(0β1),但非β-凸集的开集与闭集的例子各一个,从中点性质是否蕴涵相应凸性的角度揭示了集合的β-凸性与通常凸性之间的另一显著差异.     

紧—凸性与紧—光滑性

本文首先通过暴露集和暴露泛函的概念引入卫闭凸集的紧－严格凸、紧－强凸、紧－一致凸及紧－非常凸等概念。用对偶映射给出了Ｂａｎａｃｈ空间的两种新光滑性－紧－一致光滑与紧－非常光滑。然后特别研究了Ｂａｎａｃｈ空间的紧－非常凸与紧－非常光滑。此外还得到关于对偶映射的两个新结果。     

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。     

Banach空间中关于有界集的同时远达问题的适定性

本文研究Banach空间中关于有界集的同时远达问题的适定性,在集合的Hausdorff距离下,证明了：对自反局部一致凸Banach空间中的闭有界集K,使所有关于K的同时远达问题是适定的紧凸子集A全体在紧凸子集全体中是Gδ型集．     

锥拟凸集值映射多目标优化的解集的连通性

在一般拓扑向量空间中,本文引进了一类锥进凸（锥严格拟凸）集值映射。并在目标映射是锥拟凸（锥严格拟凸）和上半连续的条件下,利用Minkowski泛函,证明了弱有效解（有效解）集是连通的。文[9]中的结论是本文所得结果大目标映射为单值和拓扑空间为格的情况下的特例。     

A Krasnosel’skii-type result for planar sets starshaped via orthogonally convex paths

A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path. Communicated by Imre Bárány     

The integral convexity of sets and functionals in Banach spaces

By way of the Bochner integral of vector-valued functions, the integral convexity of sets and functionals and the concept of integral extreme points of sets are introduced in Banach spaces. The relations between integral convexity and convexity are mainly discussed, two integral extreme points theorems and their applications are obtained at last.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

Smooth approximation of convex functions in Banach spaces

This paper shows that every extended-real-valued lower semi-continuous proper (respectively Lipschitzian) convex function defined on an Asplund space can be represented as the point-wise limit (respectively uniform limit on every bounded set) of a sequence of Lipschitzian convex functions which are locally affine (hence, C^∞) at all points of a dense open subset; and shows an analogous for w^∗-lower semi-continuous proper (respectively Lipschitzian) convex functions defined on dual spaces whose pre-duals have the Radon-Nikodym property.     

Characterizations of strictly convex sets by the uniqueness of support points

Abstract

This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so-called regular sets, for which every extreme point is a unique support point, is introduced. Closed strictly convex sets and their intersections with some other sets are shown to belong to this class. The obtained characterizations are then applied to set-valued maps and to the separation of a convex set and a strictly convex set. Under suitable assumptions, so-called set-valued maps with path property are characterized by strictly convex images of the considered set-valued map.     

Convex envelopes generated from finitely many compact convex sets

We consider the problem of constructing the convex envelope of a lower semi-continuous function defined over a compact convex set. We formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets. In particular, we consider nonnegative functions that are products of convex and component-wise concave functions and derive closed-form expressions for the convex envelopes of a wide class of such functions. Several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques.     

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.     

向量场的积分曲线分类和链群与有向同伦不变性

利用积分曲线的极限集,对紧流形M上向量场X所确定的积分曲线进行了分类,在分类的基础上定义了向量场X的链群、极限集闭链群和极限集边缘链群,以及两个同类群,最后还引入了向量场正向同伦和负向同伦的概念,并证明了极限集是正向同伦和负向同伦不变的,链群、极限集闭链群和极限集边缘链群以及两个同类群在向量场双向同伦的情况下是同构的.     

Strict convexity of some subsets of Hankel operators

We find some extreme points in the unit ball of the set of Hankel operators and show that the unit ball of the set of compact Hankel operators is strictly convex. We use this result to show that the collection of lower triangular Toeplitz contractions is strictly convex. We also find some extreme points in certain reduced Cowen sets and discuss cases in which they are or are not strictly convex.