# Question

For which pairs $M,N$ of compact metric spaces does there exist a metric space $K$ along with isometric embeddings $i:M \to K$ and $j:N \to K$ so that the Hausdorff distance between $i(M)$ and $j(N)$ in $K$ equals the Gromov-Hausdorff distance between $M$ and $N$ exactly?

We always have $d_\text{GH}(M,N) \leq d_\text{H}(i(M),j(N))$ by definition, so it suffices to decide if there exist any constraints intrinsic to $M$ and $N$ which force the reverse inequality to also hold for some judicious choice of $i,j,K$.

# Update

Bill Johnson's nice answer appears to settle the question for compact $M$ and $N$ affirmatively: the Gromov-Hausdorff distance can always be realized by an ultraproduct construction. In light of that answer -- and also in light of the fact that I did not intend to assume compactness in the original question but stupidly wrote it down anyway -- here is a modified question:

Can the GH-distance be similarly achieved if we only assume that $M$ and $N$ are

bounded, rather than compact?

In general, I am interested in the weakest hypotheses on $M$ and $N$ which are known to guarantee embeddings into a common target space $K$ so that the Gromov-Hausdorff distance between $M$ and $N$ equals the Hausdorff distance of their isometric images in $K$. It is not clear to me that a modification of Bill's argument works when we drop compactness.

# Background

Given a compact metric space $M$ and two subspaces $A, B \subset M$, their *Hausdorff distance* -- denoted $d_\text{H}(A,B)$ -- is defined to be the smallest $\epsilon > 0$ so that $B$ is contained in $A$ thickened by $\epsilon$ and vice-versa.

The *Gromov-Hausdorff* distance $d_\text{GH}(M,N)$ between two compact metric spaces $M$ and $N$ is defined to be the infimum over triples $(i,j,K)$ of $d_\text{H}(i(M),j(N))$ where $K$ is a metric space and $i,j$ are isometric embeddings of $M, N$ into $K$. The question asks when this infimum can be explicitly realized.