## Abstract

Given compact metric spaces X and Z with Hausdorff dimension n, if there is a distance-nonincreasing onto map f : Z → X, then the Hausdorff nvolumes satisfy vol(X)≤ vol(Z) The relatively maximum volume conjecture says that if X and Zare both Alexandrov spaces and vol(X)= vol(Z), X is isometric to a gluing space produced from Z along its boundary ∂ Z and f is length-preserving. We partially verify this conjecture and give a further classification for compact Alexandrovn-spaces with relatively maximum volume in terms of a fixed radius and space of directions. We also give an elementary proof for a pointed version of the Bishop-Gromov relative volume comparison with rigidity in Alexandrov geometry.

Original language | English (US) |
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Pages (from-to) | 387-420 |

Number of pages | 34 |

Journal | Pacific Journal of Mathematics |

Volume | 259 |

Issue number | 2 |

DOIs | |

State | Published - 2012 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Alexandrov space
- Radius
- Rigidity
- Stability
- Volume