Gromov-Hausdorff distance

Definition
Suppose $$(X,d_X)$$ and $$(Y,d_Y)$$ are two defining ingredient::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 defining ingredient::Hausdorff distance between $$X$$ and $$Y$$ relative to the metric.