{"id":"https://openalex.org/W7204957236","doi":"https://doi.org/10.48550/arxiv.2608.29941","title":"Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity","display_name":"Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity","publication_year":2026,"publication_date":"2026-08-30","ids":{"openalex":"https://openalex.org/W7204957236","doi":"https://doi.org/10.48550/arxiv.2608.29941"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.29941","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29941","pdf_url":"https://arxiv.org/pdf/2608.29941","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.29941","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5018124959","display_name":"Xuan Hien Nguyen","orcid":"https://orcid.org/0000-0002-5504-2393","observed_orcids":["https://orcid.org/0000-0002-5504-2393"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nguyen, Xuan Hien","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5067198002","display_name":"Alina Stancu","orcid":"https://orcid.org/0000-0001-5883-4073","observed_orcids":["https://orcid.org/0000-0001-5883-4073"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Stancu, Alina","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9218999743461609,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9218999743461609,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.042899999767541885,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.031199999153614044,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ornstein-uhlenbeck-operator","display_name":"Ornstein-Uhlenbeck operator","score":0.9760000109672546},{"id":"https://openalex.org/keywords/torsion-function","display_name":"torsion function","score":0.8339999914169312},{"id":"https://openalex.org/keywords/convex-bodies","display_name":"convex bodies","score":0.7699999809265137},{"id":"https://openalex.org/keywords/centrally-symmetric-convex-bodies","display_name":"centrally symmetric convex bodies","score":0.7570000290870667},{"id":"https://openalex.org/keywords/mean-width","display_name":"mean width","score":0.8600000143051147},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.5040000081062317}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7652999758720398},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7210999727249146},{"id":"https://openalex.org/C77461463","wikidata":"https://www.wikidata.org/wiki/Q7827201","display_name":"Torsion (gastropod)","level":2,"score":0.6177999973297119},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6013000011444092},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5925999879837036},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5332000255584717},{"id":"https://openalex.org/C79464548","wikidata":"https://www.wikidata.org/wiki/Q464794","display_name":"Minkowski space","level":2,"score":0.5160999894142151},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5142999887466431},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.49050000309944153},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.42289999127388},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.4146000146865845},{"id":"https://openalex.org/C2991999034","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex domain","level":3,"score":0.38359999656677246},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37119999527931213},{"id":"https://openalex.org/C151756577","wikidata":"https://www.wikidata.org/wiki/Q36810","display_name":"Quadrilateral","level":3,"score":0.3138999938964844},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.28870001435279846},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.2842999994754791},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.2648000121116638},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2639000117778778},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.2621999979019165},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.26109999418258667},{"id":"https://openalex.org/C140142295","wikidata":"https://www.wikidata.org/wiki/Q617417","display_name":"Isoperimetric inequality","level":2,"score":0.25450000166893005}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.29941","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29941","pdf_url":"https://arxiv.org/pdf/2608.29941","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.29941","doi":"https://doi.org/10.48550/arxiv.2608.29941","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.29941","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.29941","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29941","pdf_url":"https://arxiv.org/pdf/2608.29941","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204957236.pdf","grobid_xml":"https://content.openalex.org/works/W7204957236.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W4401306546","https://openalex.org/W7123504341","https://openalex.org/W7135203502","https://openalex.org/W7139918456"],"related_works":["https://openalex.org/W7139918456","https://openalex.org/W4392186664","https://openalex.org/W4413303337","https://openalex.org/W7214312997","https://openalex.org/W4382023322","https://openalex.org/W2785860737","https://openalex.org/W4200529296","https://openalex.org/W4319779240","https://openalex.org/W4293369698","https://openalex.org/W3134755024"],"abstract_inverted_index":{"Let":[0,36],"$u$":[1],"be":[2,42,90],"the":[3,7,17,43,50,86,113,132,151,164,170,181,188,191,229],"torsion":[4,138,156,192],"function":[5],"for":[6,63,75,100,122,214],"Ornstein-Uhlenbeck":[8],"operator":[9],"on":[10,34,175],"a":[11,76,118,125,222],"bounded":[12],"domain":[13],"$\u03a9\\subset":[14],"\\mathbb{R}^n$,":[15],"i.e.,":[16],"solution":[18],"of":[19,78,107,180,190,201,224],"$\u0394u":[20],"-":[21],"x":[22],"\\cdot":[23],"\\nabla":[24],"u":[25],"=":[26,32,38],"-1$":[27],"in":[28,69,112],"$\u03a9$":[29],"with":[30,83,93,169],"$u":[31],"0$":[33,67],"$\\partial\u03a9$.":[35],"$T_\u03b3(\u03a9)":[37],"\\int_\u03a9u":[39],"\\,":[40],"d\u03b3$":[41],"Gaussian":[44],"torsional":[45],"rigidity.":[46],"We":[47,185],"prove":[48,186],"that":[49,187,204,211],"Brunn-Minkowski-type":[51],"inequality":[52],"$T_\u03b3((1-t)\u03a9_0":[53,143],"+":[54,59,144],"t\u03a9_1)^\u03b1":[55],"\\le":[56],"(1-t)":[57],"T_\u03b3(\u03a9_0)^\u03b1":[58],"t":[60],"T_\u03b3(\u03a9_1)^\u03b1$":[61],"fails":[62,213],"every":[64,70,215],"exponent":[65,161,172,207,217],"$\u03b1>":[66],"and":[68,105,110,139,210],"dimension":[71],"$n":[72,102],"\\ge":[73],"2$,":[74,104],"pair":[77],"convex":[79,195],"bodies":[80],"centrally":[81],"symmetric":[82,176,202],"respect":[84],"to":[85,147],"origin,":[87],"which":[88],"may":[89],"taken":[91],"smooth":[92],"positive":[94,216],"curvature.":[95],"This":[96],"answers":[97],"Conjecture":[98],"1.4":[99],"any":[101],"\\geq":[103],"Question~(Q),":[106],"Mar\u00edn":[108],"Sola":[109],"Salerno":[111],"negative.":[114],"The":[115],"mechanism":[116],"is":[117,193,221],"first-order":[119],"lower":[120],"bound":[121],"$T_\u03b3$":[123],"at":[124],"ball":[126],"$\u03a9_0$":[127],"under":[128],"Minkowski":[129,152,199],"perturbations.":[130],"When":[131],"perturbing":[133],"body":[134],"$\u03a9_1$":[135],"has":[136,154],"small":[137],"large":[140],"mean":[141],"width,":[142],"t\u03a9_1)$":[145],"increases":[146],"first":[148],"order.":[149],"Since":[150],"combination":[153],"larger":[155],"than":[157],"both":[158],"endpoints,":[159],"no":[160,205],"can":[162],"repair":[163],"inequality.":[165],"For":[166],"$n=1$,":[167],"convexity":[168,212],"optimal":[171],"$1/3$":[173],"holds":[174],"intervals":[177,226],"by":[178],"results":[179],"same":[182],"authors,":[183],"\\cite{MSS26}.":[184],"logarithm":[189],"neither":[194],"nor":[196],"concave":[197],"along":[198],"combinations":[200],"intervals,":[203],"non-zero":[206],"yields":[208],"concavity,":[209],"when":[218,228],"one":[219],"set":[220],"union":[223],"two":[225],"or":[227],"sets":[230],"are":[231],"reflected":[232],"off-center":[233],"intervals.":[234]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-02T00:00:00"}