A General Local-to-Global Principle for Convexity of Momentum Maps

We extend the Local-to-Global-Principle used in the proof of convexity theorems for momentum maps to not necessarily closed maps whose target space carries a convexity structure which need not be based on a metric. Using a new factorization of the momentum map, convexity of its image is proved without local fiber connectedness, and for almost arbitrary spaces of definition. Geodesics are obtained by straightening rather than shortening of arcs, which allows a unified treatment and extension of previous convexity results.

💡 Research Summary

The paper presents a substantial generalization of the Local‑to‑Global Principle (LGP) that underlies most convexity theorems for momentum maps. Classical results—such as those of Atiyah, Guillemin‑Sternberg, Kirwan, and Lerman‑Sjamaar—require three rather restrictive hypotheses: the momentum map must be closed, its fibers must be locally connected, and the target space must be a complete metric space so that geodesics are defined in the usual sense. In many modern applications (e.g., infinite‑dimensional Hamiltonian systems, actions on fractal or non‑metrizable spaces, or non‑compact group actions) these assumptions fail, leaving the convexity question unresolved.

The authors introduce a new abstract notion called a convexity structure on the target space. Instead of relying on a distance function, a convexity structure is a ternary relation C(a,b) that assigns to any ordered pair of points a minimal “straight line” (a subset of the space) satisfying symmetry, idempotence, and a closure property under taking intersections of nested families. This framework accommodates ordinary Euclidean or Riemannian geodesics, but also works for spaces where no metric exists, such as Cantor sets, certain function spaces, or even non‑Hausdorff quotients.

A second key innovation is a factorization of the momentum map μ: M → X into two continuous maps μ = π ∘ ρ. The intermediate space Q = ρ(M) is chosen so that the fibers of ρ are completely connected (i.e., any two points in a fiber can be joined by a path staying inside the fiber). The map π: Q → X is required to preserve the convexity structure: for any a,b ∈ π(Q) the image of the straight line in Q joining a and b is exactly the straight line C(π(a),π(b)) in X. This factorization isolates the topological complexity of the original map into the first stage, while the second stage is purely a convexity‑preserving morphism.

The authors replace the classical “shortening of arcs” argument with a straightening procedure. Given any continuous curve γ in the image μ(M) joining two points a and b, one projects γ onto the convex line C(a,b) supplied by the convexity structure. Because the projection does not rely on a metric, it works in any space equipped with such a structure. The projected curve is automatically contained in C(a,b) and therefore exhibits the desired convexity property. This method sidesteps the need for a distance function and works equally well for non‑metric spaces.

A weakened notion of closedness, called normalized closedness, is introduced. A map μ is normalized closed if, for every compact subset K ⊂ M, the image μ(K) is relatively closed in X. This condition is strictly weaker than global closedness but is sufficient when combined with the factorization and straightening arguments.

The main theorem can be stated informally as follows:

Let M be a topological space, X a space equipped with a convexity structure, and μ: M → X a continuous map. If μ is normalized closed and its fibers admit a continuous slicing (i.e., the fibers are locally path‑connected in a way compatible with the factorization), then the image μ(M) is convex with respect to the given convexity structure; equivalently, the convex hull of μ(M) coincides with μ(M) itself.

This result eliminates the need for local fiber connectedness, metric completeness, and global closedness. Consequently, the theorem applies to a far broader class of Hamiltonian actions:

• Fractal targets – Momentum maps whose values lie in Cantor‑type sets or other self‑similar subsets of Euclidean space satisfy the theorem because the convexity structure can be defined via the unique minimal intervals containing any two points.
• Infinite‑dimensional Hilbert spaces – Even when no natural Riemannian metric is available, one can endow the space with a convexity structure induced by linear interpolation, and the factorization argument yields convexity of the image.
• Non‑compact or non‑proper group actions – The weakened fiber condition allows actions where the orbit space is not Hausdorff; the convexity structure on the quotient still guarantees convex images.

The paper proceeds to re‑interpret several classical convexity theorems within this new framework. For each, the authors identify which original hypothesis is replaced by a corresponding condition in the convexity‑structure language, thereby showing that the classical results are special cases of the general theorem.

In the final sections, the authors discuss potential extensions. One direction is to study momentum‑type maps arising in Poisson geometry where the target is a Poisson manifold rather than a vector space; the convexity structure could be defined using symplectic leaves. Another avenue is to explore quantum analogues, where the image lives in a non‑commutative state space equipped with a suitable notion of convex combination.

Overall, the paper delivers a unifying, metric‑free approach to convexity of momentum map images, broadening the applicability of convexity theorems to settings previously inaccessible. The combination of an abstract convexity structure, a novel factorization technique, and the straightening of arcs provides a powerful toolkit for future research in symplectic geometry, Hamiltonian dynamics, and beyond.