### Abstract

A boundary distance representation of a Riemannian manifold with boundary (M, g, ∂M) is the set of functions {r_{x} ∈ C(∂M): x ∈ M}, where r_{x} are the distance functions to the boundary, r_{x}(z) = d(x, z), z ∈ ∂M. In this paper we study the question whether this representation determines the Riemannian manifold in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold (M, g) in the Gromov-Hausdorff topology. In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigation. As an example, for the heat equation with an unknown heat conductivity the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity.

Original language | English |
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Pages (from-to) | 135-157 |

Number of pages | 23 |

Journal | Inverse Problems and Imaging |

Volume | 1 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2007 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Modelling and Simulation
- Discrete Mathematics and Combinatorics
- Control and Optimization

### Cite this

*Inverse Problems and Imaging*,

*1*(1), 135-157. https://doi.org/10.3934/ipi.2007.1.135