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 a closed Riemannian manifold. To see that the above result also generalizes and improves an analog of the Cheeger type estimate in Alexandrov geometry in "A.D. Alexandrov spaces with curvature bounded below," 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 $n = \dim(X)$.