#### Volume 15, issue 2 (2011)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
On Gromov–Hausdorff stability in a boundary rigidity problem

### Sergei Ivanov

Geometry & Topology 15 (2011) 677–697
##### Abstract

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 ${C}^{1}$–close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance function of $M$ is ${C}^{0}$–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.

##### Keywords
boundary distance rigidity, Gromov–Hausdorff topology
Primary: 53C23
##### Publication
Received: 27 July 2010
Revised: 24 January 2011
Accepted: 22 February 2011
Published: 10 May 2011
Proposed: Dmitri Burago
Seconded: Leonid Polterovich, Jean-Pierre Otal
##### Authors
 Sergei Ivanov St.Petersburg Department Steklov Mathematical Institute RAS 27, Fontanka 191023 St Petersburg Russia http://eimi.imi.ras.ru/eng/perso/svivanov.php