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# Revisiting Tietze-Nakajima: Local and Global Convexity for Maps

Published:2010-07-06
Printed: Oct 2010
• Christina Bjorndahl,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, CANADA
• Yael Karshon,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, CANADA
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## 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.
 MSC Classifications: 53D20 - Momentum maps; symplectic reduction 52B99 - None of the above, but in this section

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