Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Metric Graph Approximations of Geodesic Spaces   A standard result in metric geometry is that every compact geodesic metric
space can be approximated arbitrarily well by finite metric graphs in the
Gromov-Hausdorff sense. It is well known that the first Betti number of the
approximating graphs may blow up as the approximation gets finer.
  In our work, given a compact geodesic metric space $X$, we define a sequence
$(\delta^X_n)_{n \geq 0}$ of non-negative real numbers by $$\delta^X_n:=\inf
\{d_{\mathrm{GH}}(X,G): G \text{ a finite metric graph, } \beta_1(G)\leq n \}
.$$
  By construction, and the above result, this is a non-increasing sequence with
limit $0$. We study this sequence and its rates of decay with $n$. We also
identify a precise relationship between the sequence and the first
Vietoris-Rips persistence barcode of $X$. Furthermore, we specifically analyze
$\delta_0^X$ and find upper and lower bounds based on hyperbolicity and other
metric invariants. As a consequence of the tools we develop, our work also
provides a Gromov-Hausdorff stability result for the Reeb construction on
geodesic metric spaces with respect to the function given by distance to a
reference point.

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The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph