Abstract
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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