We show that every dynamically convex toric domain in R⁴ admits a C¹-neighborhood U within the space of star-shaped domains of R⁴ with the following property: for any X ∈ U, every Legendrian knot in ∂ X admits a Reeb chord with distinct endpoints. A higher dimensional analog is also discussed.
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Kang et al. (2024) studied this question.
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