Gromov-Hausdorff Distances in Euclidean Spaces

Abstract

The purpose of this paper is to study the relationship between measures of dissimilarity between shapes in Euclidean space. We first concentrate on the pair Gromov-Hausdorff distance (GH) versus Hausdorff distance under the action of Euclidean isometries (EH). Then, we (1) show they are comparable in a precise sense that is not the linear behaviour one would expect and (2) explain the source of this phenomenon via explicit constructions. Finally, (3) by conveniently modifying the expression for the GH distance, we recover the EH distance. This allows us to uncover a connection that links the problem of computing GH and EH and the family of Euclidean Distance Matrix completion problems. The second pair of dissimilarity notions we study is the so called Lp-Gromov-Hausdorff distance versus the Earth Moverpsilas distance under the action of Euclidean isometries. We obtain results about comparability in this situation as well.