Bounding geometry of loops in alexandrov spaces

Nan Li University of Notre Dame Xiaochun Rong Rutgers University

Differential Geometry mathscidoc:1609.10279

Journal of Differential Geometry, 92, (1), 31-54, 2012
For a path in a compact finite dimensional Alexandrov space X with curv  , 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 , 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).
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@inproceedings{nan2012bounding,
  title={BOUNDING GEOMETRY OF LOOPS IN ALEXANDROV SPACES},
  author={Nan Li, and Xiaochun Rong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231219208913941},
  booktitle={Journal of Differential Geometry},
  volume={92},
  number={1},
  pages={31-54},
  year={2012},
}
Nan Li, and Xiaochun Rong. BOUNDING GEOMETRY OF LOOPS IN ALEXANDROV SPACES. 2012. Vol. 92. In Journal of Differential Geometry. pp.31-54. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231219208913941.
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