{"id":"https://openalex.org/W7220916929","doi":"https://doi.org/10.48550/arxiv.2610.08616","title":"On the $L$-embeddability of Lipschitz-free spaces in their biduals","display_name":"On the $L$-embeddability of Lipschitz-free spaces in their biduals","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220916929","doi":"https://doi.org/10.48550/arxiv.2610.08616"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.08616","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08616","pdf_url":"https://arxiv.org/pdf/2610.08616","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/2610.08616","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110781289","display_name":"Bruno de Mendon\u00e7a Braga","orcid":"https://orcid.org/0000-0002-3456-4002","observed_orcids":["https://orcid.org/0000-0002-3456-4002"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Braga, Bruno de Mendon\u00e7a","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5112097895","display_name":"Chris Gartland","orcid":"https://orcid.org/0009-0008-6552-5584","observed_orcids":["https://orcid.org/0009-0008-6552-5584"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gartland, Chris","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5104520044","display_name":"Gilles Lancien","orcid":"https://orcid.org/0000-0002-7901-2594","observed_orcids":["https://orcid.org/0000-0002-7901-2594"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lancien, Gilles","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5038773763","display_name":"Pavlos Motakis","orcid":"https://orcid.org/0000-0002-4395-3931","observed_orcids":["https://orcid.org/0000-0002-4395-3931"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Motakis, Pavlos","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052712578","display_name":"Eva Perneck\u00e1","orcid":"https://orcid.org/0000-0003-1600-7963","observed_orcids":["https://orcid.org/0000-0003-1600-7963"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Perneck\u00e1, Eva","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5009865100","display_name":"Thomas Schlumprecht","orcid":"https://orcid.org/0000-0002-6601-4080","observed_orcids":["https://orcid.org/0000-0002-6601-4080"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Schlumprecht, Thomas","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/T11545","display_name":"Advanced Banach Space Theory","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":1.0,"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/T10899","display_name":"Advanced Operator Algebra Research","score":0.0,"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":0.0,"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/lipschitz-free-spaces","display_name":"Lipschitz-free spaces","score":0.9800000190734863},{"id":"https://openalex.org/keywords/bidual-spaces","display_name":"bidual spaces","score":0.7799999713897705},{"id":"https://openalex.org/keywords/projection","display_name":"projection","score":0.4699999988079071},{"id":"https://openalex.org/keywords/metric-spaces","display_name":"metric spaces","score":0.7760000228881836},{"id":"https://openalex.org/keywords/compact-sets","display_name":"compact sets","score":0.6460000276565552},{"id":"https://openalex.org/keywords/banach-spaces","display_name":"Banach spaces","score":0.6700000166893005}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.691100001335144},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.5437999963760376},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5286999940872192},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.47929999232292175},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.47920000553131104},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.43369999527931213},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.41350001096725464},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4077000021934509},{"id":"https://openalex.org/C6183050","wikidata":"https://www.wikidata.org/wiki/Q3058180","display_name":"Interpolation space","level":4,"score":0.3264999985694885},{"id":"https://openalex.org/C152306953","wikidata":"https://www.wikidata.org/wiki/Q621744","display_name":"Approximation property","level":3,"score":0.3181000053882599},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.30489999055862427},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2935999929904938},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.27950000762939453},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.2786000072956085},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27810001373291016},{"id":"https://openalex.org/C196613150","wikidata":"https://www.wikidata.org/wiki/Q3966092","display_name":"Convex metric space","level":3,"score":0.2766000032424927},{"id":"https://openalex.org/C31498916","wikidata":"https://www.wikidata.org/wiki/Q583034","display_name":"Locally compact space","level":2,"score":0.27379998564720154},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2581000030040741}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.08616","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08616","pdf_url":"https://arxiv.org/pdf/2610.08616","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.2610.08616","doi":"https://doi.org/10.48550/arxiv.2610.08616","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.08616","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:2610.08616","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08616","pdf_url":"https://arxiv.org/pdf/2610.08616","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":[{"id":"https://openalex.org/G1840770272","display_name":null,"funder_award_id":"E-26/","funder_id":"https://openalex.org/F4320322749","funder_display_name":"Funda\u00e7\u00e3o Carlos Chagas Filho de Amparo \u00e0 Pesquisa do Estado do Rio de Janeiro"},{"id":"https://openalex.org/G46575062","display_name":"Composition operators and Banach spaces","funder_award_id":"ANR-24-CE40-0892","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G6421916385","display_name":"Banach Spaces: Theory and Applications","funder_award_id":"2349322","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6956711515","display_name":null,"funder_award_id":"RGPIN-2021","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G933372841","display_name":null,"funder_award_id":"E_26/204","funder_id":"https://openalex.org/F4320322749","funder_display_name":"Funda\u00e7\u00e3o Carlos Chagas Filho de Amparo \u00e0 Pesquisa do Estado do Rio de Janeiro"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320307521","display_name":"American Institute of Mathematics","ror":"https://ror.org/02a0fwv66"},{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"},{"id":"https://openalex.org/F4320322749","display_name":"Funda\u00e7\u00e3o Carlos Chagas Filho de Amparo \u00e0 Pesquisa do Estado do Rio de Janeiro","ror":"https://ror.org/03kk0s825"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220916929.pdf","grobid_xml":"https://content.openalex.org/works/W7220916929.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7221781613","https://openalex.org/W2340775323","https://openalex.org/W2963505682","https://openalex.org/W1707247767","https://openalex.org/W7222616483","https://openalex.org/W2945858065","https://openalex.org/W7147206448","https://openalex.org/W7163096951","https://openalex.org/W6966747771","https://openalex.org/W7213414766"],"abstract_inverted_index":{"A":[0],"Banach":[1],"space":[2,67],"$X$":[3,30],"is":[4,11,42,73,78,83],"$L$-embedded":[5,44],"in":[6,45],"its":[7,36,46],"bidual":[8],"if":[9],"there":[10],"a":[12,65],"projection":[13],"$P\\colon":[14],"X^{**}\\to":[15],"X$":[16],"such":[17],"that":[18],"$\\|x\\|=\\|Px\\|+\\|x-Px\\|$":[19],"for":[20,64],"all":[21],"$x\\in":[22],"X^{**}$.":[23],"We":[24],"show":[25],"that,":[26],"as":[27,29],"long":[28],"has":[31],"dimension":[32],"at":[33],"least":[34],"$2$,":[35],"Lipschitz-free":[37],"space,":[38],"denoted":[39],"by":[40],"$\\mathcal{F}(X)$,":[41],"not":[43],"bidual.":[47],"Our":[48],"methods":[49],"have":[50],"applications":[51],"to":[52,62],"the":[53,57,74],"problem":[54],"of":[55,59],"when":[56,76,81],"$L$-embeddability":[58],"$\\mathcal{F}(M)$":[60],"passes":[61],"$\\mathcal{F}(A)$":[63],"metric":[66],"$M$":[68],"and":[69],"$A\\subseteq":[70],"M$.":[71],"This":[72],"case":[75],"$A$":[77,82],"compact":[79],"or":[80],"geodesically":[84],"closed.":[85]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}
