Nearest and farthest points in spaces of curvature bounded below

Let A be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space (X,d) and x E X / A. The nearest point problem (resp. the farthest point problem) w.r.t. considered here is to find a point ao E A such that d(x,ao) = inf{d(x,a) : a E A} (resp. d(x,ao) = sup{d(x,a) : a E A})....

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Detalhes bibliográficos
Autores: Espínola García, Rafael, Li, Chong, López Acedo, Genaro
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2010
País:España
Recursos:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/182152
Acesso em linha:https://hdl.handle.net/11441/182152
https://doi.org/10.1016/j.jat.2010.02.007
Access Level:acceso abierto
Palavra-chave:Nearest and farthest points
Geodesic spaces
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spelling Nearest and farthest points in spaces of curvature bounded belowEspínola García, RafaelLi, ChongLópez Acedo, GenaroNearest and farthest pointsGeodesic spacesLet A be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space (X,d) and x E X / A. The nearest point problem (resp. the farthest point problem) w.r.t. considered here is to find a point ao E A such that d(x,ao) = inf{d(x,a) : a E A} (resp. d(x,ao) = sup{d(x,a) : a E A}). We study the well posedness of nearest point problems and farthest point problems in geodesic spaces. We show that if X is a space of curvature bounded below then the complement of the set of all points x E X for which the nearest point problem (resp. the farthest point problem) w.r.t.x is well posed is r-porous in X / A. Our results extend and/or improve some recent results about proximinality in geodesic spaces as well as the corresponding ones previously obtained in uniformly convex Banach spaces. In particular, the result regarding the nearest point problem answers affirmatively an open problem proposed by Kaewcharoen and Kirk [A. Kaewcharoen, W.A. Kirk, Proximinality in geodesic spaces, Abstr. Appl. Anal. 2006 (2006) 1–10].ElsevierAnálisis MatemáticoFQM127: Análisis Funcional no Lineal2010info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/182152https://doi.org/10.1016/j.jat.2010.02.007reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)Inglés10.1016/j.jat.2010.02.007info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1821522026-06-17T12:51:07Z
dc.title.none.fl_str_mv Nearest and farthest points in spaces of curvature bounded below
title Nearest and farthest points in spaces of curvature bounded below
spellingShingle Nearest and farthest points in spaces of curvature bounded below
Espínola García, Rafael
Nearest and farthest points
Geodesic spaces
title_short Nearest and farthest points in spaces of curvature bounded below
title_full Nearest and farthest points in spaces of curvature bounded below
title_fullStr Nearest and farthest points in spaces of curvature bounded below
title_full_unstemmed Nearest and farthest points in spaces of curvature bounded below
title_sort Nearest and farthest points in spaces of curvature bounded below
dc.creator.none.fl_str_mv Espínola García, Rafael
Li, Chong
López Acedo, Genaro
author Espínola García, Rafael
author_facet Espínola García, Rafael
Li, Chong
López Acedo, Genaro
author_role author
author2 Li, Chong
López Acedo, Genaro
author2_role author
author
dc.contributor.none.fl_str_mv Análisis Matemático
FQM127: Análisis Funcional no Lineal
dc.subject.none.fl_str_mv Nearest and farthest points
Geodesic spaces
topic Nearest and farthest points
Geodesic spaces
description Let A be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space (X,d) and x E X / A. The nearest point problem (resp. the farthest point problem) w.r.t. considered here is to find a point ao E A such that d(x,ao) = inf{d(x,a) : a E A} (resp. d(x,ao) = sup{d(x,a) : a E A}). We study the well posedness of nearest point problems and farthest point problems in geodesic spaces. We show that if X is a space of curvature bounded below then the complement of the set of all points x E X for which the nearest point problem (resp. the farthest point problem) w.r.t.x is well posed is r-porous in X / A. Our results extend and/or improve some recent results about proximinality in geodesic spaces as well as the corresponding ones previously obtained in uniformly convex Banach spaces. In particular, the result regarding the nearest point problem answers affirmatively an open problem proposed by Kaewcharoen and Kirk [A. Kaewcharoen, W.A. Kirk, Proximinality in geodesic spaces, Abstr. Appl. Anal. 2006 (2006) 1–10].
publishDate 2010
dc.date.none.fl_str_mv 2010
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/182152
https://doi.org/10.1016/j.jat.2010.02.007
url https://hdl.handle.net/11441/182152
https://doi.org/10.1016/j.jat.2010.02.007
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv 10.1016/j.jat.2010.02.007
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
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