[Paper Review] Revisiting Tietze-Nakajima - Local and Global Convexity for Maps

Yael Karshon, Christina Bjorndahl|arXiv (Cornell University)|Jan 25, 2007

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper extends the classical Tietze-Nakajima theorem to maps from topological spaces to R^n by introducing the concept of a 'convex map'—a map where any two points can be joined by a path whose image under the map is a monotone parametrization of a straight line segment. Under conditions of connectedness, properness, and local convexity and openness of the map, the paper proves that the global map is convex and open, and that its image is convex and its fibers are connected.

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 o 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 o Psi(U) is convex and open. Then the map Psi: X o Ψ(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.

Motivation & Objective

• To generalize the Tietze-Nakajima local-to-global convexity principle from subsets of R^n to continuous maps from topological spaces to R^n.
• To define and analyze a new class of maps—'convex maps'—where paths in the domain map to monotone straight-line segments in R^n.
• To establish conditions under which local convexity and openness of a map imply global convexity and openness of the map and connectedness of fibers.
• To provide an elementary, accessible proof framework suitable for advanced undergraduates, motivated by symplectic geometry applications.
• To clarify the relationship between the authors' results and prior work, particularly Birtea-Ortega-Ratiu (2007), by showing that their convexity condition is stronger but yields a cleaner, more intuitive statement.

Proposed method

• Define a map $\Psi: X \to \mathbb{R}^n$ to be convex if any two points in $X$ can be joined by a path $\gamma$ such that $\Psi \circ \gamma$ is a monotone parametrization of a line segment in $\mathbb{R}^n$.
• Assume $X$ is connected and Hausdorff, $\Psi$ is proper, and each point in $X$ has a neighborhood $U$ where $\Psi|_U$ is convex and open as a map to its image.
• Use uniform local convexity on compact sets (Lemma 3) to control path lengths and construct midpoints (Lemma 5), ensuring the existence of path-based convexity structures.
• Apply a partition of unity and local convexity to construct global paths whose images are straight lines, proving that $\Psi$ is globally convex and open.
• Leverage the fact that convex maps are locally fiber connected and have local convexity data in the broader sense of Birtea-Ortega-Ratiu (2007), but with stronger assumptions.
• Use the closedness and properness of $\Psi$ to ensure that the image is closed and convex, and that fibers are connected via path-lifting and convexity arguments.

Experimental results

Research questions

• RQ1 Can the local-to-global convexity principle of Tietze-Nakajima be extended from subsets of R^n to continuous maps from topological spaces to R^n?
• RQ2 What conditions on a map $\Psi: X \to \mathbb{R}^n$ ensure that local convexity and openness imply global convexity and connected fibers?
• RQ3 How does the new definition of 'convex map' compare to the local convexity data used in Birtea-Ortega-Ratiu (2007), and what are the implications for the strength and clarity of the results?
• RQ4 In what ways does the convex map condition simplify or strengthen the conclusions compared to prior approaches in symplectic geometry and convexity theorems?
• RQ5 Can the proof of the Atiyah-Guillemin-Sternberg convexity theorem be rederived using this elementary, path-based definition of convex maps?

Key findings

• If $X$ is connected and Hausdorff, $\Psi: X \to \mathbb{R}^n$ is proper, and each point has a neighborhood $U$ such that $\Psi|_U$ is convex and open to its image, then $\Psi$ is globally convex and open as a map to its image.
• The image $\Psi(X)$ is convex in $\mathbb{R}^n$, and the fibers $\Psi^{-1}(y)$ are connected for all $y \in \Psi(X)$.
• The proof is elementary and accessible to advanced undergraduates with basic topology, relying on path construction, uniform local convexity, and compactness arguments.
• The authors' convex map condition implies both local convexity data (in the broader sense of Birtea-Ortega-Ratiu) and local fiber connectedness, but with stronger structural assumptions.
• The inclusion map of a closed ball into $\mathbb{R}^n$ is a convex map in the authors' sense but does not satisfy the local convexity data condition in the original Birtea-Ortega-Ratiu (2007) formulation.
• The results are not essentially new in light of Birtea-Ortega-Ratiu (2007), but the authors' formulation yields cleaner, more intuitive statements and proofs.