  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 $n=\dim(X)$.
