Banach空间中的几类新可凹点（英文）  

Several New Classes of Denting Points in Banach Spaces

作　　者：王小霞 崔云安[2] 商绍强 WANG Xiaoxia;CUI Yun＇an;SHANG Shaoqiang(Ordos Insitution of Applied Technology,Ordos,Inner Mongolia,017000,P.R.China;Harbin University of Science and Technology,Harbin,Heilongjiang,150080,P.R.China;Northeast Forestry University,Harbin,Heilongjiang,150040,P.R.China)

机构地区：[1]鄂尔多斯应用技术学院,鄂尔多斯内蒙古017000 [2]哈尔滨理工大学,哈尔滨黑龙江150080 [3]东北林业大学,哈尔滨黑龙江150040

出　　处：《数学进展》2018年第6期906-912,共7页Advances in Mathematics（China）

基　　金：Supported by Science Research Project of Ordos Institution of Applied Technology(No.KYYB2017014)

摘　　要：本文在Banach空间X中引入了弱可凹点、弱局部一致可凹点、K-w强暴露点和强*可凹点几个概念.首先,证明了若单位球面S(X)上的任何一点均为单位球U(X)的弱可凹点,则X是严格凸的.其次,证明了当X为自反Banach空间时,X的单位球面S(X)上的任何一点均为单位球CU(X)的弱可凹点当且仅当X的对偶空间X~*是非常光滑的.此外,同样在X为自反Banach空间的情况下,证明了若对偶空间X*是弱局部一致光滑的,则X的单位球面S(X)上的任何一点均为单位球U(X)的弱局部一致可凹点.最后,还证明了当X为自反Banach空间时,若CU(X)为一个K-w可凹集,则任意一点x∈S(X)均为U(X)的一个K-w强暴露点,并且证明了x~*∈X~*为X~*的一个1-w~*可凹点当且仅当x~*为X~*的一个强~*可凹点.In this paper, the concepts of weak denting point, weakly locally uniformly denting point, K-w strongly exposed point and strong＊ denting point are introduced in Banach space X. First, it is proved that if every point of closed unit sphere S（X） is a weak denting point of closed unit ball U（X）, then X is strictly convex. Second, when X is a reflexive Banach space, it is proved that every point x of the closed unit sphere S（X） is a weak denting point of closed unit ball U（X） if and only if the dual space X＊ is very smooth. Moreover, in the case of a reflexive Banach space X, it is also proved that if the dual space X＊ is weakly locally uniformly smooth, then every point of the closed unit sphere S（X） is a weakly locally uniformly denting point of closed unit ball U（X）. Finally, we prove that if U（X） is a K-w dentable set, then every point x ∈ S（X） is a K-w strongly exposed point of U（X）, and x＊ ∈ X＊ is a l-w＊ denting point of X＊ if and only if x＊ is a strong＊ denting point of X＊, when X is a reflexive Banach space.

关 键 词：弱可凹点 弱局部一致可凹点 K-ω强暴露点 强^＊可凹点 

分 类 号：O177.2[理学—数学]