Geodesic equicontinuity characterization of visibility

Let ΩCN\Omega\subset\mathbb C^N be a bounded convex domain, let z0Ωz_0\in\Omega, and let Gz0\mathcal G_{z_0} be the family of all geodesic segments and geodesic rays starting from z0z_0. Geodesic equicontinuity conjecture. Then Ω\Omega is visible if and only if Gz0\mathcal G_{z_0} is uniformly equicontinuous. This would characterize visibility through the behavior of geodesics based at one interior point; the source gives no proof or counterexample.

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Primary source

Filippo Bracci and Ahmed Yekta Ökten, “Some open questions and conjectures about visibility and iteration in bounded convex domains in C^N”, arXiv:2507.19967 (2025).

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