# Persistent Homology and the Upper Box Dimension

29 Jul 2019 Schweinhart Benjamin

We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of \$\mathbb{R}^2\$ whose upper box dimension exceeds \$1.5.\$ These results are related to extremal questions about the number of persistent homology intervals of a set of \$n\$ points in a metric space...

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• METRIC GEOMETRY
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• ALGEBRAIC TOPOLOGY
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