Revisiting Tietze–Nakajima: Local and Global Convexity for Maps
Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 975-993
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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.
Bjorndahl, Christina; Karshon, Yael. Revisiting Tietze–Nakajima: Local and Global Convexity for Maps. Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 975-993. doi: 10.4153/CJM-2010-052-5
@article{10_4153_CJM_2010_052_5,
author = {Bjorndahl, Christina and Karshon, Yael},
title = {Revisiting {Tietze{\textendash}Nakajima:} {Local} and {Global} {Convexity} for {Maps}},
journal = {Canadian journal of mathematics},
pages = {975--993},
year = {2010},
volume = {62},
number = {5},
doi = {10.4153/CJM-2010-052-5},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-052-5/}
}
TY - JOUR AU - Bjorndahl, Christina AU - Karshon, Yael TI - Revisiting Tietze–Nakajima: Local and Global Convexity for Maps JO - Canadian journal of mathematics PY - 2010 SP - 975 EP - 993 VL - 62 IS - 5 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-052-5/ DO - 10.4153/CJM-2010-052-5 ID - 10_4153_CJM_2010_052_5 ER -
%0 Journal Article %A Bjorndahl, Christina %A Karshon, Yael %T Revisiting Tietze–Nakajima: Local and Global Convexity for Maps %J Canadian journal of mathematics %D 2010 %P 975-993 %V 62 %N 5 %U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-052-5/ %R 10.4153/CJM-2010-052-5 %F 10_4153_CJM_2010_052_5
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