  We address the problem of estimating intrinsic distances in a manifold from a
finite sample. We prove that the metric space defined by the sample endowed
with a computable metric known as sample Fermat distance converges a.s. in the
sense of Gromov-Hausdorff. The limiting object is the manifold itself endowed
with the population Fermat distance, an intrinsic metric that accounts for both
the geometry of the manifold and the density that produces the sample. This
result is applied to obtain sample persistence diagrams that converge towards
an intrinsic persistence diagram. We show that this method outperforms more
standard approaches based on Euclidean norm with theoretical results and
computational experiments.
