Convexity of Chebyshev Sets Revisited

Abstract

A subset C of a normed vector space V is called a Chebyshev set if every point in V admits a unique nearest point in C. In this article we give a novel proof that every Chebyshev set in n-dimensional Euclidean space is convex. This statement is sometimes referred to as the “Bunt–Motzkin Theorem.”

MSC::

• Primary 11L03
• Secondary 11B68

Acknowledgment

The authors wish to thank Dr. Tomasz Kobos for introducing us to the problem of convexity of Chebyshev sets and providing invaluable assistance and suggestions in writing and submitting this article.

Additional information

Notes on contributors

Konrad Deka

Konrad Deka received his Master’s degree in pure mathematics from Jagiellonian University, Poland, and is currently pursuing a Ph.D. at the same institution. His research interests include symbolic dynamical systems and functional analysis. Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland [email protected]

Marin Varivoda

Marin Varivoda is currently an undergraduate student in mathematics at the University of Zagreb, Croatia. In high school, he competed in the IMO and won a silver medal. His research interests include number theory and analysis. Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia [email protected]