  We start by considering the problem of estimating intrinsic distances on a
smooth submanifold. We show that minimax optimality can be obtained via a
reconstruction of the surface, and discuss the use of a particular mesh
construction -- the tangential Delaunay complex -- for that purpose. We then
turn to manifold learning and argue that a variant of Isomap where the
distances are instead computed on a reconstructed surface is minimax optimal
for the isometric variant of the problem.
