Existence of closed quasigeodesic in Alexandov spaces of some types (Q1608300)

The paper addresses generalizations of existence of closed geodesics on simply-connected compact smooth surfaces and compact Riemannian manifolds to existence of closed quasi-geodesics on hypersurfaces in \(E^n\) and certain Alexandrov spaces. The author proves that on every compact Alexandrov space \( X\) with curvature bounded below and \(\pi _1\left( X\right) \neq 0\) there exists a closed geodesic. If \(X\) is simply-connected and can be approximated by a sequence of compact Riemannian manifolds with the same lower curvature bound converging to \(X\) in the Lipschitz sense, then there exists a closed quasi-geodesic on \(X\). Another result proved in the paper is that every convex hypersurface in Euclidean space has a closed quasi-geodesic.