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CONTROLLED GEOMETRY VIA VOLUMES ON ALEXANDROV SPACES Controlled Geometry via Volumes on Alexandrov Spaces

2010
unpublished

In this thesis we recall the basic definitions and properties for Alexandrov space and describe two geometry phenomenons controlled via volume (Hausdorff measure or rough volume) conditions. (1) For a path in X ∈ Alex n (κ) (the compact n-dimensional Alexandrov spaces with curvature ≥ κ.), the sum of the length and the turning angle is bounded from below in terms of κ, n, diameter and volume of X. This generalizes a basic estimate by Cheeger on the length of a closed geodesic in closed

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