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

more »

... n manifold ([Ch]). (2) Let Σ p be the space of directions at p ∈ X and the pointed radius R = inf{r : X ⊂ B r (p)}. If X ∈ Alex n (κ), then vol(X) ≤ vol(C R κ (Σ p)), where C R κ (Σ p) is the metric R-ball at the vertex in the κ-suspension C κ (Σ p). We give an isometric classification of X ∈ Alex n (κ) whose volume achieves the maximal possible value vol(C R κ (Σ p)). We also determine homeomorphic types of such X when X is a topological manifold. These results are natural extension of K. Grove and P. Petersen's work in 1992 ([GP 92]). ii