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Revisiting Tietze–Nakajima: Local and Global Convexity for Maps

Published online by Cambridge University Press:  20 November 2018

Christina Bjorndahl  and
Yael Karshon
Christina Bjorndahl*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, CANADA
Yael Karshon*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, CANADA
*
karshon@math.toronto.edu
cjm295@cornell.edu
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Abstract

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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.

Keywords

53D2052B99
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 5 , 01 October 2010 , pp. 975 - 993
Copyright
Copyright © Canadian Mathematical Society 2010

Footnotes

*

Formerly Christina Marshall.

The first author is partially supported by an NSERC Discovery grant. The second author was partially funded by an NSERC USRA grant in the summers of 2004 and 2005.

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