About the continuity of one operation with convex compacts in finite-dimensional normed spaces

TL;DR

This paper proves that in finite-dimensional normed spaces, the intersection operation of a convex compact set with a neighborhood of another convex compact set varies continuously with the neighborhood radius, in the Hausdorff topology.

Contribution

It establishes the continuity of the intersection operation involving convex compact sets and neighborhoods in finite-dimensional normed spaces.

Findings

The intersection operation is continuous in the Hausdorff topology.

Continuity holds specifically for convex compact sets in finite-dimensional spaces.

The result was motivated by extremal network theory and has potential for generalizations.

Abstract

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.