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- Publisher Website: 10.1016/j.jat.2008.08.015
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Article: Bregman distances and Klee sets
Title | Bregman distances and Klee sets |
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Authors | |
Keywords | Farthest point Bregman distance Bregman projection Convex function Subdifferential operator Maximal monotone operator Legendre function |
Issue Date | 2009 |
Citation | Journal of Approximation Theory, 2009, v. 158, n. 2, p. 170-183 How to Cite? |
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. © 2008 Elsevier Inc. All rights reserved. |
Persistent Identifier | http://hdl.handle.net/10722/250926 |
ISSN | 2023 Impact Factor: 0.9 2023 SCImago Journal Rankings: 0.660 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Bauschke, Heinz H. | - |
dc.contributor.author | Wang, Xianfu | - |
dc.contributor.author | Ye, Jane | - |
dc.contributor.author | Yuan, Xiaoming | - |
dc.date.accessioned | 2018-02-01T01:54:06Z | - |
dc.date.available | 2018-02-01T01:54:06Z | - |
dc.date.issued | 2009 | - |
dc.identifier.citation | Journal of Approximation Theory, 2009, v. 158, n. 2, p. 170-183 | - |
dc.identifier.issn | 0021-9045 | - |
dc.identifier.uri | http://hdl.handle.net/10722/250926 | - |
dc.description.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. © 2008 Elsevier Inc. All rights reserved. | - |
dc.language | eng | - |
dc.relation.ispartof | Journal of Approximation Theory | - |
dc.subject | Farthest point | - |
dc.subject | Bregman distance | - |
dc.subject | Bregman projection | - |
dc.subject | Convex function | - |
dc.subject | Subdifferential operator | - |
dc.subject | Maximal monotone operator | - |
dc.subject | Legendre function | - |
dc.title | Bregman distances and Klee sets | - |
dc.type | Article | - |
dc.description.nature | link_to_OA_fulltext | - |
dc.identifier.doi | 10.1016/j.jat.2008.08.015 | - |
dc.identifier.scopus | eid_2-s2.0-67349163273 | - |
dc.identifier.volume | 158 | - |
dc.identifier.issue | 2 | - |
dc.identifier.spage | 170 | - |
dc.identifier.epage | 183 | - |
dc.identifier.eissn | 1096-0430 | - |
dc.identifier.isi | WOS:000266888900004 | - |
dc.identifier.issnl | 0021-9045 | - |