Let M be a compact Riemannian manifold with boundary. We show that M is Gromov-Hausdorff close to a convex Euclidean region D of the same dimension if the boundary distance function of M is C¹ -close to that of D. More generally, we prove the same result under the assumptions that the boundary distance function of M is C⁰ -close to that of D, the volumes of M and D are almost equal, and volumes of metric balls in M have a certain lower bound in terms of radius.