Quantitative isoperimetric inequalities for log-convex probability measures on the line

Quantitative isoperimetric inequalities for log-convex probability measures on the line

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\&#...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Hauptverfasser: Feo, F, Posteraro, M. R, Roberto, C
Format: Artikel
Sprache:eng
Schlagworte:
Mathematics - Differential Geometry
Mathematics - Probability
Online-Zugang:Volltext bestellen
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page
container_issue
container_start_page
container_title
container_volume
creator Feo, F
Posteraro, M. R
Roberto, C
description The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\'e). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) to any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type.
doi_str_mv 10.48550/arxiv.1401.0628
format Article
fullrecord <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_1401_0628</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1401_0628</sourcerecordid><originalsourceid>FETCH-LOGICAL-a658-b21f31912c3ff3db1de349beb58c6b8b827d2097afb9589a904df86d46422d873</originalsourceid><addsrcrecordid>eNotj0tLxDAUhbNxIaN7V5I_0JpX22Qpgy8YkIEBl-WmudFAp6lpWmb-vR11dRbnwfkIueOsVLqq2AOkU1hKrhgvWS30NfnYzzDkkCGHBWmY4ogpHDGn0NEw4PcMfcgBJ-pjon38LLo4LHiiY4oWbFjNMz0iTHNaM3Gg-QtpvxZvyJWHfsLbf92Qw_PTYfta7N5f3raPuwLqShdWcC-54aKT3ktnuUOpjEVb6a622mrROMFMA96aShswTDmva6dqJYTTjdyQ-7_ZX7B2XL9DOrcXwPYCKH8AYNZNQQ</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Quantitative isoperimetric inequalities for log-convex probability measures on the line</title><source>arXiv.org</source><creator>Feo, F ; Posteraro, M. R ; Roberto, C</creator><creatorcontrib>Feo, F ; Posteraro, M. R ; Roberto, C</creatorcontrib><description>The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\'e). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) to any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type.</description><identifier>DOI: 10.48550/arxiv.1401.0628</identifier><language>eng</language><subject>Mathematics - Differential Geometry ; Mathematics - Probability</subject><creationdate>2014-01</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1401.0628$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1401.0628$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Feo, F</creatorcontrib><creatorcontrib>Posteraro, M. R</creatorcontrib><creatorcontrib>Roberto, C</creatorcontrib><title>Quantitative isoperimetric inequalities for log-convex probability measures on the line</title><description>The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\'e). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) to any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type.</description><subject>Mathematics - Differential Geometry</subject><subject>Mathematics - Probability</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotj0tLxDAUhbNxIaN7V5I_0JpX22Qpgy8YkIEBl-WmudFAp6lpWmb-vR11dRbnwfkIueOsVLqq2AOkU1hKrhgvWS30NfnYzzDkkCGHBWmY4ogpHDGn0NEw4PcMfcgBJ-pjon38LLo4LHiiY4oWbFjNMz0iTHNaM3Gg-QtpvxZvyJWHfsLbf92Qw_PTYfta7N5f3raPuwLqShdWcC-54aKT3ktnuUOpjEVb6a622mrROMFMA96aShswTDmva6dqJYTTjdyQ-7_ZX7B2XL9DOrcXwPYCKH8AYNZNQQ</recordid><startdate>20140103</startdate><enddate>20140103</enddate><creator>Feo, F</creator><creator>Posteraro, M. R</creator><creator>Roberto, C</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20140103</creationdate><title>Quantitative isoperimetric inequalities for log-convex probability measures on the line</title><author>Feo, F ; Posteraro, M. R ; Roberto, C</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a658-b21f31912c3ff3db1de349beb58c6b8b827d2097afb9589a904df86d46422d873</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Mathematics - Differential Geometry</topic><topic>Mathematics - Probability</topic><toplevel>online_resources</toplevel><creatorcontrib>Feo, F</creatorcontrib><creatorcontrib>Posteraro, M. R</creatorcontrib><creatorcontrib>Roberto, C</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Feo, F</au><au>Posteraro, M. R</au><au>Roberto, C</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Quantitative isoperimetric inequalities for log-convex probability measures on the line</atitle><date>2014-01-03</date><risdate>2014</risdate><abstract>The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\'e). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) to any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type.</abstract><doi>10.48550/arxiv.1401.0628</doi><oa>free_for_read</oa></addata></record>
fulltext fulltext_linktorsrc
identifier DOI: 10.48550/arxiv.1401.0628
ispartof
issn
language eng
recordid cdi_arxiv_primary_1401_0628
source arXiv.org
subjects Mathematics - Differential Geometry
Mathematics - Probability
title Quantitative isoperimetric inequalities for log-convex probability measures on the line
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-25T07%3A04%3A09IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-arxiv_GOX&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Quantitative%20isoperimetric%20inequalities%20for%20log-convex%20probability%20measures%20on%20the%20line&rft.au=Feo,%20F&rft.date=2014-01-03&rft_id=info:doi/10.48550/arxiv.1401.0628&rft_dat=%3Carxiv_GOX%3E1401_0628%3C/arxiv_GOX%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true