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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 closedfatcat:5ngxwle5rvg6pi574rjmlwmtoy