(untitled) (Journal of Mathematical Study, 2025)
Let F be a closed subset in a finite dimensional Alexandrov space X with lower curvature bound. This paper shows that F is quasi-convex if and only if, for any two distinct points p,r∈F, if there is a direction at p which is more than π2 away from ⇑^r_p (the set of all directions from p to r), then the farthest direction to ⇑^r_p at p is tangent to F. This implies that F is quasi-convex if and only if the gradient curve starting from r of the distance function to p lies in F. As an application, we obtain that the fixed point set of an isometry on X is quasi-convex.