Let $M$ be a Riemannian manifold, $x$ and $y$ are two points in $M$.
Assume that $x$ is not in the cut locus of $y$. Does there exist a neighborhood $U$ of $x$ and a neighborhood $V$ of $y$ such that for every point $u$ in $U$ and for every point $v$ in $V$ we have that $u$ is not in the cut locus of $v$?
[Math] An elementary question about the cut locus
riemannian-geometry
Related Solutions
Let me continue the comments above here so I can include a figure. Here are examples of the medial axis of two different convex polygons (from my own work):
The term medial axis is used in computer science to denote the same concept as the cut locus.
Franz-Erich Wolter wrote his Ph.D. dissertation on "Cut loci in bordered and unbordered Riemannian manifolds" (Technische Universität Berlin, 1985). That might contain some useful information.
The injectivity radius for a point $x$ is the largest distance $r$ such that any geodesic starting from x is length-minimizing for at least distance $r$. So there exists at least one geodesic starting from $x$ that is not length-minimizing past distance $r$. On the other hand a point $p$ is in the cut locus of $x$ if a geodesic starting from $x$ and passing through $p$ is not length-minimizing for any point past $p$. So the injectivity radius at $x$ is the distance from $x$ to its cut locus.
Best Answer
For a unit tangent vector $u$ with footpoint $p$ let $t(u)$ be the supremum of positive numbers such that the geodesic $t\to \exp_p(tu)$ is minimizing on $[0,t(u)]$. The cut locus at $p$ is the set of points $\exp_{p}(t(u) u)$ of $M$ for which $t_u$ is finite.
A basic result is that $u\to t(u)$ defines a continuous map from the unit tangent bundle to $(0,+\infty]$ where continuity at $+\infty$ is understood in the obvious way. See e.g. Sakai's "Riemannian geometry", Proposition III.4.1.
Now coming to your question fix $x\in M$ and $y=\exp_x(su)$ for some $u=u(x,y)$ and positive number $s$. Suppose $x^\prime$, $y^\prime$ are near $x$, $y$ respectively, and write $y^\prime=\exp_{x^\prime}(s^\prime u^\prime)$. If $y$ is not in the cut locus of $x$, then $t(u)=+\infty$. So $t(v)> s$ on some neighborhood of $u$ in the unit tangent bundle. Since $s^\prime$ and $s$ are almost the same, we conclude that $t(u^\prime)> s^\prime$, i.e. $x^\prime$, $y^\prime$ are not cut points of each other.
In my view Sakai's book is the best source of information about cut and cunjugate loci.