{"id":"https://openalex.org/W7213301149","doi":"https://doi.org/10.48550/arxiv.2609.14832","title":"A constructive solution to Talagrand's Gaussian convexification problem","display_name":"A constructive solution to Talagrand's Gaussian convexification problem","publication_year":2026,"publication_date":"2026-09-13","ids":{"openalex":"https://openalex.org/W7213301149","doi":"https://doi.org/10.48550/arxiv.2609.14832"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.14832","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14832","pdf_url":"https://arxiv.org/pdf/2609.14832","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/2609.14832","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002434154","display_name":"Othmane Mazhar","orcid":"https://orcid.org/0000-0003-1521-9091","observed_orcids":["https://orcid.org/0000-0003-1521-9091"]},"institutions":[{"id":"https://openalex.org/I4405270816","display_name":"Fondation du Risque","ror":"https://ror.org/01c3q7502","country_code":"FR","type":"nonprofit","lineage":["https://openalex.org/I4405270816"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Mazhar, Othmane","raw_affiliation_strings":["Institute Louis Bachelier -Fondation du Risque"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute Louis Bachelier -Fondation du Risque","institution_ids":["https://openalex.org/I4405270816"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5002434154"],"corresponding_institution_ids":["https://openalex.org/I4405270816"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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":0.9473000168800354,"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":0.9473000168800354,"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/T11545","display_name":"Advanced Banach Space Theory","score":0.020400000736117363,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.007499999832361937,"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/minkowski-sum","display_name":"Minkowski sum","score":0.7289999723434448},{"id":"https://openalex.org/keywords/gaussian-measure","display_name":"Gaussian measure","score":0.796999990940094},{"id":"https://openalex.org/keywords/convex-sets","display_name":"convex sets","score":0.5540000200271606},{"id":"https://openalex.org/keywords/ellipsoids","display_name":"ellipsoids","score":0.7360000014305115},{"id":"https://openalex.org/keywords/balanced-sets","display_name":"balanced sets","score":0.5379999876022339}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7946000099182129},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.6504999995231628},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5835999846458435},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5049999952316284},{"id":"https://openalex.org/C57489055","wikidata":"https://www.wikidata.org/wiki/Q190046","display_name":"Ellipsoid","level":2,"score":0.4724999964237213},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.42989999055862427},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.42739999294281006},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4074000120162964},{"id":"https://openalex.org/C2777630202","wikidata":"https://www.wikidata.org/wiki/Q5527840","display_name":"Gaussian measure","level":3,"score":0.40369999408721924},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.38989999890327454},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.37720000743865967},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.3625999987125397},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.3458000123500824},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.33809998631477356},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33379998803138733},{"id":"https://openalex.org/C79464548","wikidata":"https://www.wikidata.org/wiki/Q464794","display_name":"Minkowski space","level":2,"score":0.3230000138282776},{"id":"https://openalex.org/C110202963","wikidata":"https://www.wikidata.org/wiki/Q1783542","display_name":"Convex geometry","level":5,"score":0.3199999928474426},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3133000135421753},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3037000000476837},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.28299999237060547},{"id":"https://openalex.org/C101901036","wikidata":"https://www.wikidata.org/wiki/Q2269096","display_name":"Semidefinite programming","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.2533999979496002}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.14832","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14832","pdf_url":"https://arxiv.org/pdf/2609.14832","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.2609.14832","doi":"https://doi.org/10.48550/arxiv.2609.14832","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.14832","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:2609.14832","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14832","pdf_url":"https://arxiv.org/pdf/2609.14832","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213301149.pdf","grobid_xml":"https://content.openalex.org/works/W7213301149.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W46152070","https://openalex.org/W1990770240","https://openalex.org/W1996820366","https://openalex.org/W2031442735","https://openalex.org/W2070589948","https://openalex.org/W2081780258","https://openalex.org/W2617066894","https://openalex.org/W2912522929","https://openalex.org/W4407633285","https://openalex.org/W7131849614","https://openalex.org/W7160900955","https://openalex.org/W7164134563"],"related_works":["https://openalex.org/W4407633285","https://openalex.org/W7131849614","https://openalex.org/W4241830564","https://openalex.org/W287912065","https://openalex.org/W7100348362","https://openalex.org/W46152070","https://openalex.org/W2019750125","https://openalex.org/W4283315694","https://openalex.org/W1639222083","https://openalex.org/W2515943892"],"abstract_inverted_index":{"Talagrand":[0],"asked":[1],"for":[2,78,105,141],"a":[3,6,11,16,23,37,108,138,145],"construction":[4,92,135,148],"of":[5,10,15,41],"large":[7,17],"convex":[8,39],"subset":[9],"bounded":[12],"Minkowski":[13],"sum":[14],"Gaussian":[18,42,59],"set.":[19],"We":[20,71],"first":[21],"prove":[22],"stronger":[24],"nonsymmetric":[25],"existential":[26],"statement:":[27],"if":[28,101],"$A\\subset\\mathbb{R}^n$":[29],"is":[30,93],"measurable":[31],"and":[32,62,68,75,83,114,124,144],"$\u03b3_n(A)>2/3$,":[33],"then":[34],"$A+A+A$":[35],"contains":[36],"compact":[38],"set":[40],"measure":[43],"at":[44],"least":[45],"$1/2$.":[46],"Let":[47],"now":[48],"$A$":[49],"be":[50,56],"closed":[51],"with":[52,112],"$\u03b3_n(A)\\ge7/8$,":[53],"let":[54],"$\u03a6$":[55],"the":[57,79,86,91,121,125],"standard":[58],"distribution":[60],"function,":[61],"put":[63],"$a_0=\u03a6^{-1}(3/4)$.":[64],"For":[65],"every":[66,69,106],"$\u039b>1$":[67],"$01.705510\\ldots$.":[70],"give":[72],"matching":[73],"upper":[74],"lower":[76],"bounds":[77],"optimal":[80,130],"high-measure":[81],"dilation":[82,87],"show":[84],"that":[85],"profile":[88],"used":[89],"by":[90],"sharp":[94],"among":[95],"all":[96],"symmetric":[97],"half-measure":[98],"cores.":[99],"Finally,":[100],"$\u03b3_n(A)\\ge5/6+\u03b7$,":[102],"we":[103],"construct,":[104],"$0<\\varepsilon\\le1/2$,":[107],"centered":[109],"ellipsoid":[110],"$E_\\varepsilon$":[111],"$\u03b3_n(E_\\varepsilon)\\ge1-\\varepsilon$":[113],"\\begin{equation}":[115],"\\frac{c}{\u03a6^{-1}(1-\\varepsilon/2)}\\sqrt{\\frac{\\log":[116],"n}{n}}\\,E_\\varepsilon\\subset":[117],"A+A+A.":[118],"\\end{equation}":[119],"Both":[120],"dimension":[122],"dependence":[123,126],"on":[127,150],"$\\varepsilon$":[128],"are":[129],"up":[131],"to":[132],"constants.":[133],"The":[134],"also":[136],"gives":[137],"six-summand":[139],"theorem":[140],"balanced":[142],"sets":[143],"second":[146],"finite":[147],"based":[149],"subgaussian":[151],"tests.":[152]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-16T00:00:00"}