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Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$.

Is there a convex set $K'\subset X'$ that is close to $K\subset X$?

Two spaces $X$ and $X'$ are $a$-close if there is a homeomorphism $\iota\colon X\to X'$ that changes metric by at most $a$. Two subsets $K\subset X$ and $K'\subset X'$ are $b$-close if the Hausdorff distance between $\iota(K)$ and $K'$ in $X'$ is at most $b$.

Comments.

  • I suspect that the answer is "no".
  • I do not know the answer even if $X$ is a ball in $\mathbb{R}^3$.
  • If $X=\mathbb{R}^3$, then the answer is "yes"; it is easy.
  • I also do not know the answer if $X$ is a Lobachevsky space.
  • The motivation comes from the question about the compactness of convex hull in CAT(0).
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