Approximations of 1-dimensional intrinsic persistence of geodesic spaces and their stabilityVirk, Žiga (Avtor) persistencegeodesic spaceminimal homology basisgeodesic circleRips complexA standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space $X$ carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of $X$ may be obtained by an appropriate finite sample (subset of $X$), and that persistence of any subset of $X$ is well interleaved with the persistence of $X$. It follows that the persistence of $X$ is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each $r>0$ a density $s>0$, so that for each $s$-dense sample $S \subset X$ the corresponding fundamental group (and the first homology) of the Rips complex of $S$ is isomorphic to the one of $X$, leading to an improved reconstruction result.20192020-04-18 11:38:56Neznano115224UDK: 515.14ISSN pri članku: 1139-1138DOI: 10.1007/s13163-018-0275-4COBISS_ID: 18410073OceCobissID: 16654937sl