Bounding geometry of loops in Alexandrov spaces

TL;DR

This paper establishes lower bounds on the sum of length and turning angle for loops in Alexandrov spaces, generalizing Cheeger's estimates from Riemannian geometry to a broader geometric setting.

Contribution

It introduces bounds on loop invariants in Alexandrov spaces, extending Cheeger's estimates and relating Hausdorff measure to volume in this context.

Findings

Sum of length and turning angle bounded below by geometric parameters

Generalizes Cheeger's estimate to Alexandrov spaces

Shows proportionality between Hausdorff measure and volume in certain subsets

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 $n=\dim(X)$.