For questions involving the Hausdorff distance (also known as the Hausdorff metric) between closed, bounded, non-empty subsets of normed linear spaces (or more generally, any metric space).
Given two non-empty, closed, bounded subsets $A, B$ of a metric space $(M, d)$, the Hausdorff distance (or Hausdorff metric) $d_H$ is defined by: \begin{align*} d_H(A, B) &= \max\left\{\sup_{a \in A}d(a, B), \sup_{b \in B}d(b, A)\right\}\\ &= \max\left\{\sup_{a \in A}\inf_{b \in B} d(a, b), \sup_{b \in B}\inf_{a \in A} d(a, b)\right\}. \end{align*} It is frequently applied to normed linear spaces, including $\Bbb{R}^n$ or $\Bbb{C}^n$, where $d(a, b) = \|a - b\|$, and is sometimes restricted to the case where $A$ and $B$ are compact.
The Hausdorff distance forms a metric on the space of non-empty, closed bounded subsets of $M$, and as such, is a measure of similarity between such sets.
