Revisiting Tietze-Nakajima: Local and Global Convexity for Maps

Christina Bjorndahl, Yael Karshon
2010 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
A theorem of Tietze and Nakajima, from 1928, asserts that if a subset X of R n is closed, connected, and locally convex, then it is convex [Ti, N]. There are many generalizations of this "local to global convexity" phenomenon in the literature; a partial list is [BF, C, Ka, KW, Kl, SSV, S, Ta]. This paper contains an analogous "local to global convexity" theorem when the inclusion map of X to R n is replaced by a map from a topological space X to R n that satisfies certain local properties: We
more » ... cal properties: We define a map Ψ : X → R n to be convex if any two points in X can be connected by a path γ whose composition with Ψ parametrizes a straight line segment in R n and this parametrization is monotone along the segment. See Definition 7. We show that, if X is connected and Hausdorff, Ψ is proper, and each point has a neighbourhood U such that Ψ| U is convex and open as a map to its image, then Ψ is convex and open as a map to its image. We deduce that the image of Ψ is convex and the level sets of Ψ are connected. See Theorems 15 and 10. Our motivation comes from the Condevaux-Dazord-Molino proof [CDM, HNP] of the Atiyah-Guillemin-Sternberg convexity theorem in symplectic geometry [At, GS1] . See section 7. This paper is the result of an undergraduate research project that spanned over the years [2004][2005][2006]. The senior author takes the blame for the delay in publication after posting our arXiv eprint. While preparing this paper we learned of the paper [BOR1] by Birtea, Ortega, and Ratiu, which achieves similar goals. In section 8 we discuss relationships between our results and theirs. After [BOR1], our results are not essentially new, but our notion of "convex map" gives elegant statements, and our proofs are so elementary that they are accessible to undergraduate students with basic topology background. * formerly Christina Marshall. 1 lemma holds with any ε < min J 1 L J where J runs over the subsets of {1, . . . , n} for which {α j | j ∈ J} are linearly independent.
doi:10.4153/cjm-2010-052-5 fatcat:srs3cc24wzcm3bomu4lc57acj4