局部非锥凸性
Locally nonconical convexity
摘要：
凸集的结构条件有一个层次。在本文中，我们研究了最近定义的一个条件[3，8，9]称为局部非锥凸（简称LNC）。很容易证明每一个严格凸集都是LNC的，半空间和这两种类型的集合的有限交集也是如此，但更多的集合是LNC的。例如，每一个znc（非原子向量值测度的值域）都是LNC（推论34）。然而，不存在无限维紧LNC集（定理23）。LNC的概念起源于搜索连续的部分，本文件显示了它如何导致自然（和建设性）在各种情况下的连续部分。设Q是R^n中的紧凸集，T是R^n到R^m的线性映射.我们证明了（定理1）Q是LNC当且仅当任何这样的T到Q的限制是Q到T（Q）的开映射。这意味着如果Q是LNC，那么任何这样的T都有从T（Q）映射到Q的连续截面（即存在T的连续右逆），并且实际上可以以各种自然的方式构造性地定义连续截面（定理3，推论4和定理5）。如果Q是严格凸的，且T不是1-1，则我们可以构造取Q边界值的连续截面（定理6）。当我们放弃紧致性时，自然会考虑希尔伯特空间X的一个闭的凸的LNC子集Q，它可以是无限维的。在这种情况下，我们必须假设T是左Fredholm，即具有闭值域和有限维核的有界线性映射。然后我们可以证明类似于上一段中提到的结果（定理16-20）。我们还证明了T（Q）是LNC（定理25）。
英文摘要：
There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as are half-spaces and finite intersections of sets of either of these types, but many more sets are LNC. For instance, every zonoid (the range of a nonatomic vector-valued measure) is LNC (Corollary 34). However, there are no infinite-dimensional compact LNC sets (Theorem 23). The LNC concept originated in a search for continuous sections, and the present paper shows how it leads naturally (and constructively) to continuous sections in a variety of situations. Let Q be a compact, convex set in R^n, and let T be a linear map from R^n into R^m. We show (Theorem 1) that Q is LNC if and only if the restriction of any such T to Q is an open map of Q onto T(Q). This implies that if Q is LNC, then any such T has continuous sections (i.e. there are continuous right inverses of T) that map from T(Q) to Q, and in fact it is possible to define continuous sections constructively in various natural ways (Theorem 3, Corollary 4, and Theorem 5). If Q is strictly convex and T is not 1-1, we can construct continuous sections which take values in the boundary of Q (Theorem 6). When we give up compactness it is natural to consider a closed, convex, LNC subset Q of a Hilbert space X which may be infinite-dimensional. In this case we must assume that T is left Fredholm, i.e. a bounded linear map with closed range and finite-dimensional kernel. We can then prove results analogous to those mentioned in the last paragraph (Theorems 16-20). We also prove that T(Q) is LNC (Theorem 25).
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