Revisiting Tietze-Nakajima: Local and Global Convexity for Maps

http://dx.doi.org/10.4153/CJM-2010-052-5
Canad. J. Math. 62(2010), 975-993

  Published:2010-07-06
 Printed: Oct 2010

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.