Optimal mutual location of compact sets in spaces endowed with Euclidean invariant Gromov-Hausdorff metric
The Euclidean invariant Gromov-Hausdorff distance between two compact subsets $A$, $B$ of $\mathbb{R}^n$ is defined as the infimum over all Hausdorff distances from $A$ to an image of $B$ under a member of the group $G$ of all orientation preserving motions of $\mathbb{R}^n$. This gives a metric on the set of $G$-orbits of compact subsets of $\mathbb{R}^n$. Two compact subsets of $\mathbb{R}^n$ are said to be in optimal mutual location if the infimum as above is attained when the motion is the identity. \par In the paper under review, the author works through several examples of compact sets in optimal mutual location, including when one of them is a singleton, both are 2-point sets, or both are straight line segments. In each of these cases the Chebyshev centers of the compact sets coincide. The author further provides an example consisting of a 3-point set and a 2-point set such that neither the Chebyshev centers coincide nor their optimal mutual location is unique. Moreover, given positive numbers it is shown that there are three compact sets such that any two are in optimal mutual location and such that their Euclidean invariant Gromov-Hausdorff distances match the three given numbers. \par In the last section of the paper, the author also deals with orbits of compact sets under the group of orientation preserving similarities of $\mathbb{R}^n$. It is shown that for two compact sets in the same orbit under this larger group their Euclidean invariant Gromov-Hausdorff distance is the absolute value of the difference of their Chebyshev radii and that their Chebyshev centers coincide if they are in optimal mutual location.
scientific article; zbMATH DE number 7023560