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.

Keywords

Quasi-convex subset / Alexandrov space / extremal subset / gradient curve

Acknowledgement

The first author and the third author are supported by National Natural Science Founda-tion of China (Grant No. 12371050).

References

[1] Alexander S, Kapovitch V, Petrunin A. Alexandrov geometry: foundations. Arxiv.org/abs/1903.08539.

[2] Burago Y, Gromov M, Perel′man G. A.D. Aleksandrov spaces with curvature bounded below. Uspeckhi Mat. Nank, 1992, 47(2): 3-51.

[3] Perel′man G, Petrunin A. Extremal subsets in Alexandrov spaces and the generalized Liber-man Theorem, Algebra i Analiz, 1993, 5(1); English transl. in St. Petersberg Math. J, 1994, 5: 215-227.

[4] Perel′man G, Petrunin A. Quasigeodesics and Gradient Curves in Alexandrov spaces. www.math.psu.edu/Petrunin/papers/.

[5] Petrunin A. Semiconcave functions in Alexandrov geometry, Surveys in differential geometry. Surv. Differ. Geom., 2007, XI: 137-201.

[6] Su X, Sun H, Wang Y. Quasi-convex subsets in Alexandrov spaces with lower curvature bound. Front. Math. China, 2022, 17(6): 1063-1082