Bounding geometry of loops in Alexandrov spaces

Abstract

For a path in a compact finite dimensional Alexandrov space $X$ with curv $\ge \kappa$, 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 closed Riemannian manifold ([Ch], [GP1,2]). To see that the above result also generalizes and improves an analogous 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 $n=\dim(X)$.