Curves in hyperspaces obtained by intersection of r-neighborhoods with a fixed subset

2025/12/06 by Galstyan, Arsen, Tuzhilin, Alexey
#46B20 #46B50 #51F99 #52A07 #52A40 #52A41 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2512.06327

Abstract

The present paper generalizes the result from one of the papers by Galstyan. Namely, we consider two nonempty subsets A and B of a metric space X, and construct one-parametric family F_r of subsets obtained by intersection between B and closed r-neighborhood of A, where r is bigger than the infimum distance between the sets A and B. In the case where B is compact, we show that this intersection, considered as a mapping, is right semicontinuously on r in the topology generated by Hausdorff distance. Moreover, if A and B are convex subsets of a normed space X, then we prove that F_r depends continuously on r in such topology if and only if the Hausdorff distance between different sets F_r is finite. We also show that for normed spaces X of dimension 2 or less, the latter condition is automatically fulfilled. For dimension 3 and hence for bigger ones, we construct an example in which the Hausdorff distance between different F_r is always infinite.

Citations

• Boundary stability in the Fermat--Steiner problem in hyperspaces over finite-dimensional normed spaces
• About the continuity of one operation with convex compacts in finite-dimensional normed spaces

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