# mg.metric geometry – Does the Hausdoff dimension characterise CAT(0) spaces having some bilipschitz balls?

It is well-known that the Hausdorff dimension is invariant under bi-Lipschitz mappings. I would be interested in a specific converse of this invariance. Let $$X$$ and $$Y$$ be two CAT(0) spaces having the same Hausdorff dimension. Can we find two open balls $$B_1 subset X$$ and $$B_2subset Y$$ and a bi-Lipschitz surjective mapping $$fcolon B_1to B_2$$?

Does it help if both spaces are geodesically complete?