Hausdorff distance

The notion of Hausdorff distance can be used to measure the distance between sets. It constitutes a practical theoretical tool to quantify the quality of an approximation.

\[ d_H(\mathcal{X}, \mathcal{Y}) = \max \left( \sup_{x ∈ \mathcal{X}}\inf_{y ∈ \mathcal{Y}} \Vert x - y \Vert, \sup_{y ∈ \mathcal{Y}}\inf_{x ∈ \mathcal{X}} \Vert x - y \Vert \right)\]