We introduce a fractal dimension for a metric space based on the persistent homology of  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 R^2 whose upper box dimension exceeds 1.5  .
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 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 .
R
based on
defined in terms of
A
extremal
A
. These results are related to extremal questions about the number of persistent homology intervals of a set of n points in a metric space