Revisiting Tietze-nakajima – Local and Global Convexity for Maps

A theorem of Tietze and Nakamija [Ti, N], from 1928, asserts that if a subset X of R is closed, connected, and locally convex, then it is convex. There are many generalizations of this “local to global convexity” phenomenon in the literature. See, e.g., [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 is replaced by a map from a topological space X to R that satisfies certain local properties: We define a map Ψ: X → R 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, and this parametrization is monotone along the segment. See Definition 6. We show that, if X is connected and Hausdorff, Ψ is proper, and each point has a neighborhood U such that Φ|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 11 and 9. Our motivation comes from the Condevaux-Dazord-Molino proof of the Atiyah-GuilleminSternberg convexity theorem in symplectic geometry. See [CDM, HNP, At, GS]. See section 7. While preparing this paper we learned of the paper [BOR], by Birtea, Ortega, and Ratiu, which has similar goals. See Theorem 2.28 of [BOR]. Strictly speaking, our results neither follow from nor imply those of [BOR]. For example, the inclusion map of a ball into R is convex in our sense but it doesn’t “have local convexity data” in the sense of [BOR, Def. 2.7]. On the other hand, a priori it is not clear whether a map that “has local convexity data and is locally fiber connected” [BOR, Definitions 2.7 and 2.15] is convex in our sense. Also, [BOR] allow the target space to be an infinite dimensional vector space; see [BOR, Theorem 2.31]. Finally, [BOR] work out numerous applications in symplectic geometry. Our paper has the advantage that it is completely elementary and requires minimal background.