We show that reconstructing a curve in $\mathbb{R}^d$ for $d\geq 2$ 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 $\mathbb{R}^2$ and $\frac{1}{3}$-samples in $\mathbb{R}^d$ for $d\geq 3$. 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.

翻译：我们显示,从0.66美元的样本中重建一个以$mathbb{R ⁇ d$计算的曲线,从$d\geq 2美元重建一个以$d\geq$为单位的曲线,总是有可能使用类似于古典NNN-Crust算法的算法。以前,只有0.47美元的样本($mathbb{R ⁇ 2$和$\frac{1 ⁇ 3}美元)才可能使用这种算法。此外,我们还表明,从0.72美元的样本中重建一个曲线并非总有独特的方法;这以前只知道是用$$美元标法。我们还将这种非独有的结果扩大到所有更高维度的超表层。