A compact Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined (up to an isometry) by the distances between the boundary points; and is said to be a minimal filling if it has the least volume among all compact Riemannian manifolds with the same boundary and the same or greater boundary distances. I will discuss the following result of a recent joint work with S. Ivanov: Euclidean regions with Riemannian metrics sufficiently close to a Euclidean one are minimal fillings and boundary rigid.