Asymptotic estimates for the largest volume ratio of a convex body

Autores
Galicer, Daniel Eric; Merzbacher, Diego Mariano; Pinasco, Damian
Año de publicación
2021
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
The largest volume ratio of a given convex body K ⊂ Rn is defined as lvr(K) := sup L⊂Rn vr(K, L), where the sup runs over all the convex bodies L. We prove the following sharp lower bound: c √n ≤ lvr(K), for every body K (where c > 0 is an absolute constant). This result improves the former best known lower bound, of order n/log log(n). We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that lvr(K) behaves as the square root of the dimension of the ambient space in the following cases: if K is the unit ball of an unitary invariant norm in Rd×d (e.g., the unit ball of the p-Schatten class Sd p for any 1 ≤ p ≤ ∞), if K is the unit ball of the full/symmetric tensor product of p-spaces endowed with the projective or injective norm, or if K is unconditional.
Fil: Galicer, Daniel Eric. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Merzbacher, Diego Mariano. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
Materia
VOLUME RATIO
RANDOM POLYTOPES
UNCONDITIONAL CONVEX BODIES
SCHATTEN CLASES
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/154848
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/154848
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Asymptotic estimates for the largest volume ratio of a convex bodyGalicer, Daniel EricMerzbacher, Diego MarianoPinasco, DamianVOLUME RATIORANDOM POLYTOPESUNCONDITIONAL CONVEX BODIESSCHATTEN CLASEShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1The largest volume ratio of a given convex body K ⊂ Rn is defined as lvr(K) := sup L⊂Rn vr(K, L), where the sup runs over all the convex bodies L. We prove the following sharp lower bound: c √n ≤ lvr(K), for every body K (where c > 0 is an absolute constant). This result improves the former best known lower bound, of order n/log log(n). We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that lvr(K) behaves as the square root of the dimension of the ambient space in the following cases: if K is the unit ball of an unitary invariant norm in Rd×d (e.g., the unit ball of the p-Schatten class Sd p for any 1 ≤ p ≤ ∞), if K is the unit ball of the full/symmetric tensor product of p-spaces endowed with the projective or injective norm, or if K is unconditional.Fil: Galicer, Daniel Eric. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Merzbacher, Diego Mariano. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaEuropean Mathematical Society2021-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/154848Galicer, Daniel Eric; Merzbacher, Diego Mariano; Pinasco, Damian; Asymptotic estimates for the largest volume ratio of a convex body; European Mathematical Society; Revista Matematica Iberoamericana; 37; 6; 3-2021; 1-260213-2230CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=37&iss=6&rank=9info:eu-repo/semantics/altIdentifier/doi/10.4171/rmi/1263info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1901.00771info:eu-repo/semantics/altIdentifier/doi/10.48550/arXiv.1901.00771info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-10T13:01:22Zoai:ri.conicet.gov.ar:11336/154848instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-10 13:01:22.702CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Asymptotic estimates for the largest volume ratio of a convex body
title Asymptotic estimates for the largest volume ratio of a convex body
spellingShingle Asymptotic estimates for the largest volume ratio of a convex body
Galicer, Daniel Eric
VOLUME RATIO
RANDOM POLYTOPES
UNCONDITIONAL CONVEX BODIES
SCHATTEN CLASES
title_short Asymptotic estimates for the largest volume ratio of a convex body
title_full Asymptotic estimates for the largest volume ratio of a convex body
title_fullStr Asymptotic estimates for the largest volume ratio of a convex body
title_full_unstemmed Asymptotic estimates for the largest volume ratio of a convex body
title_sort Asymptotic estimates for the largest volume ratio of a convex body
dc.creator.none.fl_str_mv Galicer, Daniel Eric
Merzbacher, Diego Mariano
Pinasco, Damian
author Galicer, Daniel Eric
author_facet Galicer, Daniel Eric
Merzbacher, Diego Mariano
Pinasco, Damian
author_role author
author2 Merzbacher, Diego Mariano
Pinasco, Damian
author2_role author
author
dc.subject.none.fl_str_mv VOLUME RATIO
RANDOM POLYTOPES
UNCONDITIONAL CONVEX BODIES
SCHATTEN CLASES
topic VOLUME RATIO
RANDOM POLYTOPES
UNCONDITIONAL CONVEX BODIES
SCHATTEN CLASES
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv The largest volume ratio of a given convex body K ⊂ Rn is defined as lvr(K) := sup L⊂Rn vr(K, L), where the sup runs over all the convex bodies L. We prove the following sharp lower bound: c √n ≤ lvr(K), for every body K (where c > 0 is an absolute constant). This result improves the former best known lower bound, of order n/log log(n). We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that lvr(K) behaves as the square root of the dimension of the ambient space in the following cases: if K is the unit ball of an unitary invariant norm in Rd×d (e.g., the unit ball of the p-Schatten class Sd p for any 1 ≤ p ≤ ∞), if K is the unit ball of the full/symmetric tensor product of p-spaces endowed with the projective or injective norm, or if K is unconditional.
Fil: Galicer, Daniel Eric. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Merzbacher, Diego Mariano. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
description The largest volume ratio of a given convex body K ⊂ Rn is defined as lvr(K) := sup L⊂Rn vr(K, L), where the sup runs over all the convex bodies L. We prove the following sharp lower bound: c √n ≤ lvr(K), for every body K (where c > 0 is an absolute constant). This result improves the former best known lower bound, of order n/log log(n). We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that lvr(K) behaves as the square root of the dimension of the ambient space in the following cases: if K is the unit ball of an unitary invariant norm in Rd×d (e.g., the unit ball of the p-Schatten class Sd p for any 1 ≤ p ≤ ∞), if K is the unit ball of the full/symmetric tensor product of p-spaces endowed with the projective or injective norm, or if K is unconditional.
publishDate 2021
dc.date.none.fl_str_mv 2021-03
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/154848
Galicer, Daniel Eric; Merzbacher, Diego Mariano; Pinasco, Damian; Asymptotic estimates for the largest volume ratio of a convex body; European Mathematical Society; Revista Matematica Iberoamericana; 37; 6; 3-2021; 1-26
0213-2230
CONICET Digital
CONICET
url http://hdl.handle.net/11336/154848
identifier_str_mv Galicer, Daniel Eric; Merzbacher, Diego Mariano; Pinasco, Damian; Asymptotic estimates for the largest volume ratio of a convex body; European Mathematical Society; Revista Matematica Iberoamericana; 37; 6; 3-2021; 1-26
0213-2230
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=37&iss=6&rank=9
info:eu-repo/semantics/altIdentifier/doi/10.4171/rmi/1263
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1901.00771
info:eu-repo/semantics/altIdentifier/doi/10.48550/arXiv.1901.00771
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv European Mathematical Society
publisher.none.fl_str_mv European Mathematical Society
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.993085

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