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- Publisher Website: 10.1016/j.jat.2008.08.014
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Article: Bregman distances and Chebyshev sets
Title | Bregman distances and Chebyshev sets |
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Authors | |
Keywords | Chebyshev set with respect to a Bregman distance Bregman projection Legendre function Maximal monotone operator Nearest point Subdifferential operators Bregman distance |
Issue Date | 2009 |
Citation | Journal of Approximation Theory, 2009, v. 159, n. 1, p. 3-25 How to Cite? |
Abstract | A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given. © 2008 Elsevier Inc. All rights reserved. |
Persistent Identifier | http://hdl.handle.net/10722/250925 |
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 | 2018-02-01T01:54:05Z | - |
dc.date.available | 2018-02-01T01:54:05Z | - |
dc.date.issued | 2009 | - |
dc.identifier.citation | Journal of Approximation Theory, 2009, v. 159, n. 1, p. 3-25 | - |
dc.identifier.issn | 0021-9045 | - |
dc.identifier.uri | http://hdl.handle.net/10722/250925 | - |
dc.description.abstract | A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given. © 2008 Elsevier Inc. All rights reserved. | - |
dc.language | eng | - |
dc.relation.ispartof | Journal of Approximation Theory | - |
dc.subject | Chebyshev set with respect to a Bregman distance | - |
dc.subject | Bregman projection | - |
dc.subject | Legendre function | - |
dc.subject | Maximal monotone operator | - |
dc.subject | Nearest point | - |
dc.subject | Subdifferential operators | - |
dc.subject | Bregman distance | - |
dc.title | Bregman distances and Chebyshev sets | - |
dc.type | Article | - |
dc.description.nature | link_to_OA_fulltext | - |
dc.identifier.doi | 10.1016/j.jat.2008.08.014 | - |
dc.identifier.scopus | eid_2-s2.0-67349154463 | - |
dc.identifier.volume | 159 | - |
dc.identifier.issue | 1 | - |
dc.identifier.spage | 3 | - |
dc.identifier.epage | 25 | - |
dc.identifier.eissn | 1096-0430 | - |
dc.identifier.isi | WOS:000267132100002 | - |
dc.identifier.issnl | 0021-9045 | - |