Approximating Gromov-Hausdorff Distance in Euclidean Space

Understand

The Gromov-Hausdorff distance $(d_{GH})$ proves to be a useful distance measure between shapes.

• In order to approximate $d_{GH}$ for compact subsets $X,Y\subset\mathbb{R}^d$, we look into its relationship with $d_{H,iso}$, the infimum Hausdorff distance under Euclidean isometries.
• As already known for dimension $d\geq 2$, the $d_{H,iso}$ cannot be bounded above by a constant factor times $d_{GH}$.
• For $d=1$, however, we prove that $d_{H,iso}\leq\frac{5}{4}d_{GH}$.