Gromov-Hausdorff distance

Definition

Suppose $(X,d_{X})$ and $(Y,d_{Y})$ are two compact metric spaces. The Gromov-Hausdorff distance between $X$ and $Y$ is defined as follows.

Let $Z$ be the disjoint union of $X$ and $Y$. Call a metric $d_{Z}$ on $Z$ admissible if the restriction of $d_{Z}$ to $X$ is $d_{X}$ and the restriction of $d_{Z}$ to $Y$ is $d_{Y}$. Then, the Gromov-Hausdorff distance between $X$ and $Y$ is the infimum, over all admissible metrics on $Z$, of the Hausdorff distance between $X$ and $Y$ relative to the metric.