On one metric in the space of nonempty closed subsets of $\mathbb R^n$

In the work, there is presented a new metric in the space $\operatorname{clos}(\mathbb R^n)$ of all nonempty closed (not necessarily bounded) subsets of $\mathbb R^n$. The convergence of sets in this metric is equivalent to convergence in the Hausdorff metric of the intersections of the given sets with the balls of any positive radius centered at zero united then with the corresponding spheres. It is proved that, with respect to the metric considered, the space $\operatorname{clos}(\mathbb R^n)$ is complete, and its subspace of nonempty closed convex subsets of $\mathbb R^n$ is closed. There are also derived the conditions that guarantee the equivalence of convergence in this metric to convergence in the Hausdorff metric, and to convergence in the Hausdorff–Bebutov metric. The results obtained can be applied to studying control problems and differential inclusions.