## Abstract

For a path in a compact finite dimensional Alexandrov space X with curv ≥ k, the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of k, the dimension, diameter, and Hausdorff measure of X. This generalizes a basic estimate of Cheeger on the length of a closed geodesic in a closed Riemannian manifold ([Ch], [GP1, 2]). To see that the above result also generalizes and improves an analog of the Cheeger type estimate in Alexandrov geometry in [BGP], we show that for a class of subsets of X, the n-dimensional Hausdorff measure and rough volume are proportional by a constant depending on n = dim(X).

Original language | English (US) |
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Pages (from-to) | 31-54 |

Number of pages | 24 |

Journal | Journal of Differential Geometry |

Volume | 92 |

Issue number | 1 |

DOIs | |

State | Published - 2012 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology