Revisiting Tietze–Nakajima: Local and Global Convexity for Maps
Revisiting Tietze–Nakajima: Local and Global Convexity for Maps
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重温 Tietze-Nakajima：地图的局部和全局凸性
DOI：
10.4153/cjm-2010-052-5
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Yael Karshon
中科院分区：
文献类型：
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作者：
Christina Bjorndahl;Yael Karshon
Tietze和Nakajima（1928）的一个定理断言，如果${{\mathbb{R}}^{n}}$的子集$X$是闭的、连通的、局部凸的，那么它是凸的。当$X$到${{\mathbb{R}}^{n}}$的包含映射被从拓扑空间$X$到${{\mathbb{R}}^{n}}$的映射满足某些局部性质时，我们给出了一个类似的“局部到全局凸性”定理。我们的动机来自于辛几何中对Atiyah-Guillemin-Sternberg凸性定理的Condevaux-Dazord-Molino证明。
Abstract A theorem of Tietze and Nakajima, from 1928, asserts that if a subset $X$ of ${{\mathbb{R}}^{n}}$ is closed, connected, and locally convex, then it is convex. We give an analogous “local to global convexity” theorem when the inclusion map of $X$ to ${{\mathbb{R}}^{n}}$ is replaced by a map from a topological space $X$ to ${{\mathbb{R}}^{n}}$ that satisfies certain local properties. Our motivation comes from the Condevaux–Dazord–Molino proof of the Atiyah–Guillemin–Sternberg convexity theorem in symplectic geometry.