The Crust algorithm by Amenta, Bern, and Eppstein computes a polygonal reconstruction of a smooth curve $C$ without boundary from a discrete set of sample points $S$. It is known that if $S$ is an a $\epsilon$-sample of $C$ with $\epsilon < \frac{1}{5}$, then the Crust algorithm computes the correct polygonal reconstruction, in the sense that it correctly connects all pairs of sample points that are consecutive along $C$. (Here, $\epsilon$-sample is defined with respect to the medial axis of $C$, as in Tamal Dey, "Curve and Surface Reconstruction", [http://www.csun.edu/~ctoth/Handbook/chap35.pdf](http://www.csun.edu/%7Ectoth/Handbook/chap35.pdf) .)


In practice, the Crust algorithm often computes correct polygonal reconstructions from samples that are less dense than a $\frac{1}{5}$-sample. What is the smallest $\epsilon > 0$ for which there exists a smooth curve $C$ without boundary and an $\epsilon$-sample $S$ of $C$ such that the Crust algorithm computes an *incorrect* polygonal reconstruction of $C$? Clearly such $\epsilon$ must be at least $\frac{1}{5}$.



What is the most dense sample for which the Crust algorithm returns an incorrect polygonal reconstruction?
In 2022, Håvard Bakke Bjerkevik gave a 0.72-sample of a particular curve for which reconstruction is not possible. Moreover, Bjerkevik showed that curve reconstruction is always possible from a 0.66-sample. This work appears in "Tighter Bounds for Reconstruction from ϵ-samples," <https://doi.org/10.4230/LIPIcs.SoCG.2022.9>.
It seems the question of curve reconstruction from an ϵ-sample is still open for ϵ-samples with 0.66 < ϵ < 0.72. 


