Convex hulls of quasi-convex sets in proper CAT(0) spaces

Question

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded neighborhood of $A$?

I'm taking the definition that $A$ is quasi-convex if there is a $d$ such that every geodesic between two points in $A$ lies in the $d$ neighborhood of $A$.

If there is some obvious counterexample to this that I've missed, then I can restrict it even further to the case that I'm most interested in. We may also assume that every geodesic between points in $A$ is $b$-contracting. i.e. All balls (regardless of size) disjoint from the geodesic project onto it as a set of diameter less than $b$.

Answer 1

There are counter-examples to this even among 3-dimensional Hadamard Riemannian manifolds of curvature $\le -1$. They are due to Ancona:

Ancona, Alano, Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature, Rev. Mat. Iberoam. 10, No. 1, 189-220 (1994). ZBL0804.58056.

He constructs a 3-dimensional Hadamard manifold $X$ of sectional curvature $\le -1$ and a point $\xi$ in the ideal boundary of $X$ such that for every neighborhood $U$ of $\xi$ in the visual compactification of $X$, the convex hull of $U\cap X$ is the entire $X$. It is easy to construct neighborhoods $U$ such that $U\cap X$ is quasiconvex and is not Hausdorff-close to $X$. For instance, take $x\in X$, $y\in x\xi$ (the ray from $x$ to $\xi$), and an open ball $B=B(y,r)$ not containing $x$. Then take the set $V$ which is the union of rays from $x$ through points of $B$, and add to this union the points at infinity that these rays represent. The result is $U$. Quasiconvexity of $V=U\cap X$ follows from Gromov-hyperbolicity of $X$.

More examples can be found in the follow-up paper

Hummel, Christoph; Lang, Urs; Schroeder, Viktor, Convex hulls in singular spaces of negative curvature, Ann. Global Anal. Geom. 18, No. 2, 191-204 (2000). ZBL0993.53012.