The Strange World of the Hausdorff Metric Geometry

I. Introduction

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.

The work presented in this web paper is the result of the investigations and efforts of my research groups in the Grand Valley State University Research Experiences for Undergraduates program, These groups have involved extremely talented students from across the country:

· Dominic Braun (then at the University of North Carolina - Asheville, graduate school at the University of Virginia, current location unknown) in our 2000 REU,

· John Mayberry (then at the University of California at Fullerton, graduate school at the University of Southern California, current location in California, but I don’t remember exactly where) and Audrey Powers (now Malagon, then at Agnes Scott College, graduate school at Emory University, now at Mercer university) in the 2002 program,

· Christopher Bay (then at Truman State University, graduate school at SUNY Stony Brook, I knew of his current location but can’t find the information now – where are you Chris?) and Amber Lembcke (Concordia College) in 2003,

· Kris Lund (then at GVSU, now working at Foremost Insurance) and Patrick Sigmon (then at Wake Forest University, now at Harvard law School) in 2004,

· Chantel Blackburn (then at Andrews University, now at the University of Arizona) and Alex Zupan (then at Gustavus Adolphus College, now at the University of Iowa) in 2005,

· Lisa Morales (then at California State Polytechnic University Pomona, now at the University of California - Riverside) and Dan Schultheis (then at the University of Washington, now at the University of California – San Diego) in 2006,

· Katrina Honigs (then at Grinnnell College, now at the University of California at Berkeley) and Vincent Martinez (then at the College of New Jersey, now at Indiana University) in 2007, and

· David Montague (University of Michigan) in 2008.

I want to express my appreciation for all the creativity and hard work they brought to this research.

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.

Last updated on May 4, 2010.

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).