BOUNDING GEOMETRY OF LOOPS IN ALEXANDROV SPACES

For a path in a compact finite dimensional Alexandrov space X with curv >= kappa, 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 kappa, 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).