弱凸集与非凸模
Weakly convex sets and modulus of nonconvexity
摘要：
考虑弱凸集的定义，它是Vial意义下的弱凸集和Clarke意义下的近光滑集的推广，从希尔伯特空间的情形推广到一类具有二阶凸性凸性模的巴拿赫空间.利用弱凸集的新定义，在给定的非凸模下，证明了一个新的收缩定理，得到了两个连续集值映射（其中一个具有非凸象）的交的连续性的新结果和选择分裂问题的新的正解.我们还研究了新定义与邻近光滑集和光滑集定义之间的关系。
英文摘要：
We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.
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