Bounding geometry of loops in Alexandrov spaces (Q1931310)

Let \(X\) be an Alexandrov space with \(\operatorname{curv} \geq\kappa,\) that is, \(X\) is a length metric space such that each point has a neighborhood in which any geodesic triangle looks fatter than a comparison triangle in the 2-dimensional space form \(S_{\kappa}^{2}\) of constant curvature \(\kappa.\) Let \(\gamma:[0,1]\rightarrow X\) be a continuous curve. Given a partition, \(P:0=t_{1}<\dots<t_{m+1}=1\) with partition size \(|P|=\delta,\) let \(p_{i} =\gamma(t_{i}),\) and let \(\gamma_{m}\) denote an \(m\)-broken geodesic, i.e., \(\gamma_{m}|_{[t_{i},t_{i+1}]}=[p_{i}p_{i+1}]\) is a minimal geodesic joining \(p_{i}\) and \(p_{i+1}.\) Let \(\theta_{i}=\pi-\measuredangle p_{i-1}p_{i} p_{i+1}.\) In particular, \(\theta_{1}=\pi-\measuredangle p_{m+1}p_{1}p_{2}\) if \(p_{m+1}=p_{1}\) (the loop case) and \(\theta_{1}=0\) otherwise. Let \(\Theta _{P}(\gamma)=\sum_{i=1}^{m}\theta_{i}.\) Then the number \(\Theta(\gamma )=\lim_{\delta\rightarrow0}\sup_{|P|=\delta}\{\Theta_{P}(\gamma)\}\) is called the ``turning angle'' of \(\gamma.\) For an Alexandrov space, the two basic geometric invariants are the length and the turning angle of a path \(\gamma\) which measures the closeness of \(\gamma\) to being a geodesic. The main result proven by the authors is that the sum of the two invariants of any loop is bounded from below in terms of \(\kappa\), the dimension, the diameter, and the Hausdorff measure of \(X.\) This generalizes a basic estimate of Cheeger on the length of a closed geodesic in a closed Riemannian manifold. The paper contains also many corollaries that are interesting in themselves.