关于凸集的两个必充条件

Two necessary and sufficient Conditions of convex set

本文是在[1]中P.10的引理和定理的基础上提出的凸集的两个必充条件。文中的定理2的必要性也是[1]中P.10定理的推广。定义1 设A为线性空间X的一个子集。A关于X的柱心记为cor(A)。它是由A中所有满足下列条件的点a所构成: 对任一yex\{a},存在bε(a、y)使[a,b](?)A。如果A=cor(A),则称A为代数开。如果x(?)cor(A)且x(?)cor(X\A)。

This paper is proposed on the basis of the lemma in Holmes K.R.Geometrial functional analysis and its applications. The two necessary and sufficient conolitions are1. Let A is a closed subset of a linear space X. Tben A is convex if and only if for any p∈ cos(A) we have cor(A)=U{[p, x]/x∈A}2. Let A is a balanced closed subset of a lineal space X. Then A is absorbent convex if and only if for any p∈cor(A), we have cot (A)=U{[p, x]/x∈lina (A)}