积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.

The Integral Convexity and Their Applications

In this paper, via Bochner integral of vector-valued functions, the authors introduce the concepts of integral convex sets and integral convex functionals and integral extremal points of sets in Banach spaces. The authors mainly show that every finite dimensional convex set and every open or closed convex set are integral convex; every lower semi-continuous convex functional and every upper semi-continuous convex functional defined on a open convex set are integral convex; every nonempty compact sets have integral extremal points; the integral extremal points set is equal to the extremal points set for every compact convex set. Two applications of integral convexity are obtained at last.