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Sampling $C^1$-submanifolds of $\mathbb{H}^n$

Volume 168 / 2022

Aleksander Antasik Colloquium Mathematicum 168 (2022), 211-228 MSC: Primary 53A35; Secondary 55P10, 53Z50. DOI: 10.4064/cm8447-6-2021 Published online: 6 December 2021


We consider the problem of reconstructing the topology of a manifold $M$ from a finite sample of its points. For a submanifold of the Euclidean $n$-space Niyogi, Smale and Weinberger (2008) have shown that the $\varepsilon $-neighbourhood of a sufficiently dense sample is homotopy equivalent to $M$. We show a similar result for $C^1$-submanifolds of $\mathbb {H}^n$. The density bound we obtain turns out to be, in general case, less efficient. We construct an example showing that it is almost as good as it can be. For a closed, connected, $1$-dimensional submanifold there is a better bound.


  • Aleksander AntasikUniversity of Wrocław
    Faculty of Mathematics and Computer Science
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland

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