Revisiting Tietze–Nakajima: Local and Global Convexity for Maps

Bjorndahl, Christina ; Karshon, Yael

Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 975-993

Résumé

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.

Détail

DOI : 10.4153/CJM-2010-052-5

Mots-clés : 53D20, 52B99

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