Abstract view
Revisiting TietzeNakajima: Local and Global Convexity for Maps


Published:20100706
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
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 CondevauxDazordMolino proof
of the AtiyahGuilleminSternberg convexity theorem in symplectic geometry.