Year: 2025
Author: Xiaole Su, Hongwei Sun, Yusheng Wang
Journal of Mathematical Study, Vol. 58 (2025), Iss. 1 : pp. 22–37
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v58n1.25.02
Journal of Mathematical Study, Vol. 58 (2025), Iss. 1 : pp. 22–37
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Quasi-convex subset Alexandrov space extremal subset gradient curve.