Optimal isometries for a pair of compact convex subsets of $\mathbb R^n$
Volume 84 / 2009
Banach Center Publications 84 (2009), 111-120
MSC: 52A20, 52A99, 41A65, 41A99.
DOI: 10.4064/bc84-0-7
Abstract
In 1989 R. Arnold proved that for every pair $(A,B)$ of compact convex subsets of $\mathbb R$ there is an Euclidean isometry optimal with respect to $L_2$ metric and if $f_0$ is such an isometry, then the Steiner points of $f_0(A)$ and $B$ coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for $L_p$ metrics for all $p \ge 2$ and the symmetric difference metric.
