Revisiting Tietze-Nakajima - Local and Global Convexity for Maps

Abstract

A theorem of Tietze and Nakamija, from 1928, asserts that if a subset X of 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 R^n is replaced by a map from a topological space X to R^n that satisfies certain local properties. We say that a map from a topological space to R^n is convex if every two points in the space can be connected by a path whose composition with the map is a weakly monotone parametrization of a straight line segment. Let X be a connected Hausdorff topological space, let T be a convex subset of R^n, and let Psi: X \to T be a continuous proper map. Suppose that every point in X is contained in an open set U such that the map Psi|_U: U \to Psi(U) is convex and open. Then the map Psi: X \to \Psi(X) is convex and open. Consequently, its image is convex and its level sets are connected. Our motivation comes from the Condevaux-Dazord-Molino proof of the Atiyah-Guillemin-Sternberg convexity theorem in symplectic geometry.