Hausdorff Distance And Lipschitz Map

In this paper, we first introduce the notions of Hausdorff distance and Gromov-Hausdorff distance. We prove, by example, that the maps from a triangle in the Euclidean plane to its centroid (incenter, circumcenter, ortho-center) are not Lipschitz maps, where the distance between two triangles is the Hausdorff distance in the Euclidean plane. On the other hand, we prove that the map form a circle or a rectangle in the Euclidean plane to its centroid is a Lipschitz map, the Lipschitz constant of the former is 1 and the Lipschitz constant of the latter is no more than 2.