  In this paper, we introduce quasi-convex subsets in Alxandrov spaces with
lower curvature bound, which include not only all closed convex subsets without
boundary but also all extremal subsets. Moreover, we explore several essential
properties of such kind of subsets including a generalized Liberman theorem. It
turns out that the quasi-convex subset is a nice and fundamental concept to
illustrate the similarities and differences between Riemannian manifolds and
Alxandrov spaces with lower curvature bound.
