几乎凸的性质及应用

Properties of Nearly Convexity and Their Applications

首先定义一个集值映射,λ:S→2(0,1),λ(S)=｛λ∈(0,1|■x,y∈S■λx+(1-λ)y∈S｝.并证明了以下结果:λ(S)≠φ■cl(λ(S))=[0,1]co(S)■cl(S);2)∩η∈гλ(Sη)≠φ■cl(∩η∈гλ(Sη))=[0,1].基于以上结果,给出了向量函数、集值映射等函数的拟凸性在半连续下的特性。

Let set-valued function A：λ：S→2^（0,1）,λ（S）={λ∈（0,1）｜for any x,y∈S→λx＋（1-λ）y∈S}. We have proved that λ（s）≠φ→cl（λ（S））=[0,1],co（S）belong to cl（S） and ∩η∈Гλ（Sη）≠φ→cl（∩ηГλ（Sη））=[0,1]. Based on these prop- erties, we give characterizations of quasiconvexity for vector-valued functions and set-valued maps under semi-continuity.