  In this paper, we study the deformation of the intersection of one compact
set with a closed neighborhood of another compact set by changing the radius of
this neighborhood. It is shown that in finite-dimensional normed spaces, in the
case when both compact sets are non-empty convex subsets, such an operation is
continuous in the topology generated by the Hausdorff metric. The question of
the continuous dependence of the described intersection on the radius of the
neighborhood arose as a by-product of the development of the theory of extremal
networks. However, it turned out to be interesting in itself, suggesting
various generalizations. Therefore, it was decided to publish it separately.
