In this talk (joint with A Katsuda, M Lassas and M Taylor)
we formulate geometric conditions on Riemannian manifolds with
boundary which guarantee their (pre)compactness in the
Gromov-Hausdorff topology. We study the metric properties of the
closure of this set. We show that the Gel'fand inverse boundary
spectral problem is uniquely solvable in this class and prove a
corresponding result on relative stability.