Very Convex Banach Spaces

ＶｅｒｙＣｏｎｖｅｘＢａｎａｃｈＳｐａｃｅｓＴｅｇｕｓｉ（特古斯）Ｓｕｙａｌａｔｕ（苏雅拉图）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＩｎｎｅｒＭｏｎｇｏｌｉａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｈｕｈｈｏｔ，０１００２２）ＬｉＹｏｎｇｊｉｎ...     

A Proximal-Type Method for Convex Vector Optimization Problem in Banach Spaces

In this paper, we consider a convex vector optimization problem of finding weak Pareto optimal solutions from a uniformly convex and uniformly smooth Banach space to a real Banach space, with respect to the partial order induced by a closed, convex, and pointed cone with a nonempty interior. We introduce a vector-valued proximal-type method based on the Lyapunov functional, carry out convergent analysis on this method, and prove that any sequence generated by the method weakly converges to a weak Pareto optimal solution of the vector optimization problem under some mild conditions. Our results improve and generalize some known results.     

More on P-Stable Convex Sets in Banach Spaces

We study the asymptotic behavior and limit distributions for sums S _n =b_n ^-1 _i=1 ⁿ _i,where _i, i 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes S _n(t) =b_n ^-1 _i=1 ^[nt] _i, t[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where _i are segments, the limit of S _n is proved to be countable zonotope. Furthermore, if B = R ^d , the singularity of distributions of two countable zonotopes Y_{p 1}, ₁,Y_{p 2}, ₂, corresponding to values of exponents p ₁, p ₂ and spectral measures ₁, ₂, is proved if either p ₁ p ₂ or _{1 2}; (iv) Some new simple estimates of parameters of stable laws in R ^d , based on these results are suggested.     

Coarse Embedding into Uniformly Convex Banach Spaces

In this paper, the author studies the coarse embedding into uniformly convex Banach spaces. The author proves that the property of coarse embedding into Banach spaces can be preserved under taking the union of the metric spaces under certain conditions. As an application, for a group G strongly relatively hyperbolic to a subgroup H,the author proves that B(n) = {g ∈ G | |g|S∪H≤ n} admits a coarse embedding into a uniformly convex Banach space if H does.     

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．     

The Geometry of Convex Transitive Banach Spaces

Throughout this paper, X will denote a Banach space, S=S(X)and B=B(X) will be the unit sphere and the closed unit ballof X, respectively, and G=G(X) will stand for the group of allsurjective linear isometries on X. Unless explicitly statedotherwise, all Banach spaces will be assumed to be real. Nevertheless,by passing to real structures, the results remain true for complexspaces. 1991 Mathematics Subject Classification 46B04, 46B10,46B22.     

Banach空间上凸泛函的某些对偶性质

本语文对Ｂａｎａｃｈ空间上的凸泛函建立了若干对偶性质,通过对偶的手段,建立了非常简洁的凸泛函极小化序列弱收敛的特征定理。     

Banach空间中的β扰动优化

通过在Banach空间中引入一种新拓扑来证明其上的β扰动优化定理成立。     

Metric Entropy of Convex Hulls in Banach Spaces

The paper presents diverse methods for estimating the coveringnumber of a precompact subset of a Banach space when the entropyof the set of its extremal points is already known. In the caseof a Hilbert space, the Gelfand diameters of the subset arealso estimated.     

On k-Strictly Convex and k-Very Smooth Banach Spaces

Ｏｎｋ－ＳｔｒｉｃｔｌｙＣｏｎｖｅｘａｎｄｋ－ＶｅｒｙＳｍｏｏｔｈＢａｎａｃｈＳｐａｃｅｓＺｈａｎｇＺｉｈｏｕ（张子厚）（ＨｕａｉｎａｎＴｅｌｅｃｉｓｉｏｎＵｎｉｖｅｒｓｉｔｙ,Ｈｕａｉｎａｎ,Ａｎｈｕｉ,２３２００１）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓ...     

B-空间上凸函数的若干指数性质

本文引入且用次梯度集该划Ｂａｎａｃｈ空间上凸函数的上、下指数,得到共轭凸函数上、下指数间的完整共轭关系、正齐次凸函数是Ｍｉｎｋｏｖｓｋｉ规函数的幂等结果,已有的一维相关结果是本文的特例。     

若干近凸空间和近光滑空间的注记

指出了关于近-强凸(近-非常凸、局部近-一致光滑)的两种不同形式的定义实质上是等价的,从而统一了有关文献中的一些结果.     

Banach空间中的弱凸集和W-太阳集

本文在光滑Banach空间的框架下,引进弱凸集和W-太阳集的概念,研究它们性质,并给出了在逼近问题中的应用.     

关于Banach空间中凸泛函的广义次梯度不等式

本文在前人^[1,2]的基础之上，以凸泛函的次梯度不等式为工具，将Jensen不等式推广到Banach空间中的凸泛函，导出了Banach空间中的Bochner积分型的广义Jensen不等式，给出其在Banach空间概率论中某些应用，从而推广了文献[3—6]的工作．     

Uniform Embeddings of Free Products into Uniformly Convex Banach Spaces

Let A and B be countable discrete groups, and let = A * B betheir free product. We show that if A and B are uniformly embeddableinto a uniformly convex Banach space, then so is . 2000 MathematicsSubject Classification 46L89, 20F65.     

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.     

Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X ^* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with $g \leqslant f$ on X is generically Fréchet differentiable. If, in addition, $\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty$ , then X is an Asplund space.     

一致Banach空间中非扩张映象的弱收敛定理

设犈是一致凸Ｂａｎａｃｈ空间，满足Ｏｐｉａｌ条件或具有Ｆｒｅｃｈｅｔ可微范数，犆是犈的非空闭凸子集，且犜：犆→犆是非扩张映象．又设对任何初始数据狓１ ∈犆，序列｛狓狀｝由下列修改了的Ｉｓｈｉｋａｗａ迭代程序生成：狓狀＋１ ＝狋狀犜狀（狊狀犜狀狓狀＋ （１－狊狀）狓狀）＋ （１－狋狀）狓狀，　狀≥１， （Ｉ）其中，数列｛狋狀｝与｛狊狀｝满足下列条件（ｉ）和（ｉｉ）之一：（ｉ）狋狀∈ ［犪，犫］且狊狀∈ ［０，犫］；（ｉｉ）狋狀∈ ［犪，１］且狊狀∈ ［犪，犫］，这里，常数犪，犫满足０＜犪≤犫＜１．作者证明了，犜有不动点的充要条件是，｛狓狀｝ 弱收敛且｛‖狓狀－犜狓狀‖｝收敛到０．而且，由此即知，若犜有不动点，则｛狓狀｝弱收敛到犜的一个不动点．     

Mazur Intersection Properties and Differentiability of Convex Functions in Banach Spaces

It is proved that the dual of a Banach space with the Mazurintersection property is almost weak* Asplund. Analogously,the predual of a dual space with the weak* Mazur intersectionproperty is almost Asplund. Through the use of these arguments,it is found that, in particular, almost all (in the Baire sense)equivalent norms on l₁() and l() are Fréchet differentiableon a dense G subset. Necessary conditions for Mazur intersectionproperties in terms of convex sets satisfying a Krein–Milmantype condition are also discussed. It is also shown that, ifa Banach space has the Mazur intersection property, then everysubspace of countable codimension can be equivalently renormedto satisfy this property.