On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space

We study the size of the sets of gradients of bump functions on the Hilbert space l(2), and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in l(2) can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert...

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Detalhes bibliográficos
Autores: Azagra Rueda, Daniel, Jiménez Sevilla, María Del Mar
Tipo de documento: artigo
Data de publicação:2002
País:España
Recursos:Universidad Complutense de Madrid (UCM)
Repositório:Docta Complutense
Idioma:inglês
OAI Identifier:oai:docta.ucm.es:20.500.14352/57098
Acesso em linha:https://hdl.handle.net/20.500.14352/57098
Access Level:Acceso aberto
Palavra-chave:517.98
Gradient
Bump function
Starlike body
Rolles Theorem
Análisis funcional y teoría de operadores
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spelling On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert spaceAzagra Rueda, DanielJiménez Sevilla, María Del Mar517.98GradientBump functionStarlike bodyRolles TheoremAnálisis funcional y teoría de operadoresWe study the size of the sets of gradients of bump functions on the Hilbert space l(2), and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in l(2) can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space l(2) can be uniformly approximated by C-1 smooth Lipschitz functions psi so that the cones generated by the ranges of its derivatives psi'(l(2)) have empty interior. This implies that there are C-1 smooth Lipschitz bumps in l(2) so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct C-1-smooth bounded starlike bodies A subset of l(2), which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to A have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.Société Mathématique de FranceUniversidad Complutense de Madrid20022002-01-0120022002-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/57098reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/570982026-06-02T12:44:21Z
dc.title.none.fl_str_mv On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
title On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
spellingShingle On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
Azagra Rueda, Daniel
517.98
Gradient
Bump function
Starlike body
Rolles Theorem
Análisis funcional y teoría de operadores
title_short On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
title_full On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
title_fullStr On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
title_full_unstemmed On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
title_sort On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
dc.creator.none.fl_str_mv Azagra Rueda, Daniel
Jiménez Sevilla, María Del Mar
author Azagra Rueda, Daniel
author_facet Azagra Rueda, Daniel
Jiménez Sevilla, María Del Mar
author_role author
author2 Jiménez Sevilla, María Del Mar
author2_role author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 517.98
Gradient
Bump function
Starlike body
Rolles Theorem
Análisis funcional y teoría de operadores
topic 517.98
Gradient
Bump function
Starlike body
Rolles Theorem
Análisis funcional y teoría de operadores
description We study the size of the sets of gradients of bump functions on the Hilbert space l(2), and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in l(2) can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space l(2) can be uniformly approximated by C-1 smooth Lipschitz functions psi so that the cones generated by the ranges of its derivatives psi'(l(2)) have empty interior. This implies that there are C-1 smooth Lipschitz bumps in l(2) so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct C-1-smooth bounded starlike bodies A subset of l(2), which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to A have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.
publishDate 2002
dc.date.none.fl_str_mv 2002
2002-01-01
2002
2002-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/57098
url https://hdl.handle.net/20.500.14352/57098
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Société Mathématique de France
publisher.none.fl_str_mv Société Mathématique de France
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,301629

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