## Abstract

In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then-analogously to the Euclidean distance case-every Klee set must be a singleton. We provide two proofs of this result, based on Monotone Operator Theory and on Nonsmooth Analysis. The latter approach leads to results that complement the work by Hiriart-Urruty on the Euclidean case.

Original language | English |
---|---|

Pages (from-to) | 170-183 |

Number of pages | 14 |

Journal | Journal of Approximation Theory |

Volume | 158 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2009 |

## Scopus Subject Areas

- Analysis
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics

## User-Defined Keywords

- Bregman distance
- Bregman projection
- Convex function
- Farthest point
- Legendre function
- Maximal monotone operator
- Subdifferential operator