TY - JOUR
T1 - Quasi-Convex Subsets and the Farthest Direction in Alexandrov Spaces with Lower Curvature Bound
AU - Su , Xiaole
AU - Sun , Hongwei
AU - Wang , Yusheng
JO - Journal of Mathematical Study
VL - 1
SP - 22
EP - 37
PY - 2025
DA - 2025/03
SN - 58
DO - http://doi.org/10.4208/jms.v58n1.25.02
UR - https://global-sci.org/intro/article_detail/jms/23936.html
KW - Quasi-convex subset, Alexandrov space, extremal subset, gradient curve.
AB - <p style="text-align: justify;">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.</p>