The Strange World of the Hausdorff Metric Geometry

I. Introduction

During the Grand Valley State university Research Experiences for Undergraduates program, my groups have investigated the geometry of the Hausdorff metric. The Hausdorff metric was first introduced by Felix Hausdorff (born November 8, 1868 in Breslau, Germany) in the early 20^th century. This metric was needed as a way to measure the distance between compact sets. In addition to his work with this metric, Hausdorff was also active in partially ordered sets, metric spaces, and Hausdorff dimension. In 1942, after being forced to retire because of his Jewish heritage, Felix Hausdorff chose to take his own life rather than be placed in a concentration camp.

In this informal paper, we will introduce the Hausdorff metric, discuss some of its applications, and review the results obtained by the GVSU REU research groups on the geometry this metric imposes on the relevant space. The list of references is at the end, with links to the appropriate web sites when possible. Throughout the paper, Java applets will appear to illustrate or highlight the ideas presented in the paper. These pages are a work in progress. As such, if you have any comments, suggestions, links or information to add, please let me know.

If you have problems viewing any of the applets on these pages, please try this.

Acknowledgements

This material is based upon work supported by the National Science Foundation under REU Grants DMS-0451254, DMS-0137264, and DMS-9820221. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF).