Quasi-Convex Subsets and the Farthest Direction in Alexandrov Spaces with Lower Curvature Bound

Abstract

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 $\frac{π}{2}$ 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.