Geometric reconstruction and persistence methods

In the present work we reconstruct the homotopy type of an unknown Euclidean subspace from a known sample of data. We carry out such reconstruction through generalized Čech complexes, by choosing radii which are less or equal than the reach of the subspace and by applying the Nerve Lemma. We also approach the reconstruction of a geodesic subspace through its convexity radius and a dense enough sample. Afterwards, we obtain homology and homotopy groups in terms of persistences, together with interleavings and isomorphisms between them. We conclude studying the reconstruction of a particular subspace that has reach equal to zero, where our results cannot be applied