Tighter Bounds for Reconstruction from ε-samples

12/07/2021

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by   Håvard Bakke Bjerkevik, et al.

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We show that reconstructing a curve in ℝ^d for d≥ 2 from a 0.66-sample is always possible using an algorithm similar to the classical NN-Crust algorithm. Previously, this was only known to be possible for 0.47-samples in ℝ^2 and 1/3-samples in ℝ^d for d≥ 3. In addition, we show that there is not always a unique way to reconstruct a curve from a 0.72-sample; this was previously only known for 1-samples. We also extend this non-uniqueness result to hypersurfaces in all higher dimensions.