Revisiting Tietze-Nakajima - Local and Global Convexity for Maps

TL;DR

This paper extends a classical convexity theorem by establishing a local-to-global convexity result for maps from topological spaces to Euclidean space, with applications in symplectic geometry.

Contribution

It introduces a new local-to-global convexity theorem for continuous maps from topological spaces to R^n, generalizing the classical convexity theorem of Tietze and Nakamija.

Findings

The map from the topological space to R^n is convex and open under local convexity conditions.

The image of the map is convex and its level sets are connected.

The theorem has applications in symplectic geometry, specifically in the proof of the convexity theorem.

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.