{"id":"https://openalex.org/W7212265147","doi":"https://doi.org/10.48550/arxiv.2609.11080","title":"A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps","display_name":"A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212265147","doi":"https://doi.org/10.48550/arxiv.2609.11080"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.11080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11080","pdf_url":"https://arxiv.org/pdf/2609.11080","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/2609.11080","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006407834","display_name":"Geunsu Choi","orcid":"https://orcid.org/0000-0002-4321-1524","observed_orcids":["https://orcid.org/0000-0002-4321-1524"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Choi, Geunsu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5006407834"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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":0.5932999849319458,"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":0.5932999849319458,"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":0.3707999885082245,"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/T11049","display_name":"Advanced Harmonic Analysis Research","score":0.010599999688565731,"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/bi-lipschitz-embeddings","display_name":"bi-Lipschitz embeddings","score":0.8889999985694885},{"id":"https://openalex.org/keywords/density","display_name":"density","score":0.609000027179718},{"id":"https://openalex.org/keywords/hausdorff-measure","display_name":"Hausdorff measure","score":0.9570000171661377},{"id":"https://openalex.org/keywords/complete-metric-spaces","display_name":"complete metric spaces","score":0.7820000052452087}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8701000213623047},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.8468999862670898},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.6886000037193298},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5856000185012817},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.5788999795913696},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.5440999865531921},{"id":"https://openalex.org/C13474197","wikidata":"https://www.wikidata.org/wiki/Q1158876","display_name":"Lipschitz domain","level":3,"score":0.5358999967575073},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.5098000168800354},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.49459999799728394},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4641000032424927},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4456000030040741},{"id":"https://openalex.org/C145084290","wikidata":"https://www.wikidata.org/wiki/Q2713824","display_name":"Metric map","level":4,"score":0.4122999906539917},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.3718000054359436},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3653999865055084},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.34209999442100525},{"id":"https://openalex.org/C12871628","wikidata":"https://www.wikidata.org/wiki/Q7598372","display_name":"Standard probability space","level":2,"score":0.3197000026702881},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3190999925136566},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.3077000081539154},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.2994999885559082},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.2915000021457672},{"id":"https://openalex.org/C84496893","wikidata":"https://www.wikidata.org/wiki/Q1135782","display_name":"Urysohn and completely Hausdorff spaces","level":4,"score":0.28700000047683716},{"id":"https://openalex.org/C119238805","wikidata":"https://www.wikidata.org/wiki/Q229102","display_name":"Surjective function","level":2,"score":0.27399998903274536},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.25690001249313354}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.11080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11080","pdf_url":"https://arxiv.org/pdf/2609.11080","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.2609.11080","doi":"https://doi.org/10.48550/arxiv.2609.11080","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.11080","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:2609.11080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11080","pdf_url":"https://arxiv.org/pdf/2609.11080","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":[{"id":"https://openalex.org/F4320320671","display_name":"National Research Foundation","ror":"https://ror.org/05s0g1g46"},{"id":"https://openalex.org/F4320322120","display_name":"National Research Foundation of Korea","ror":"https://ror.org/013aysd81"},{"id":"https://openalex.org/F4320328359","display_name":"Ministry of Science and ICT, South Korea","ror":"https://ror.org/01wpjm123"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212265147.pdf","grobid_xml":"https://content.openalex.org/works/W7212265147.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7167905527"],"related_works":["https://openalex.org/W7024771974","https://openalex.org/W4389325879","https://openalex.org/W4416538066","https://openalex.org/W2956393637","https://openalex.org/W2905888823","https://openalex.org/W2955834111","https://openalex.org/W2517686101","https://openalex.org/W4415276274","https://openalex.org/W2865543895","https://openalex.org/W4362637604"],"abstract_inverted_index":{"We":[0],"completely":[1],"characterize":[2],"the":[3,30,45,90,115,123],"denseness":[4,46],"of":[5,20,32,58,89,93,97,106,109],"strongly":[6,14,33,98],"minimum-attaining":[7,34,99],"Lipschitz":[8,16,35,100],"functions,":[9,17],"a":[10,39,56,69,82],"minimum":[11],"analogue":[12],"for":[13],"norm-attaining":[15],"in":[18,95],"terms":[19,96],"bi-Lipschitz":[21,53,70,104],"embeddings.":[22],"More":[23],"precisely,":[24],"our":[25],"main":[26,116],"result":[27,117],"shows":[28],"that":[29,114],"set":[31],"functions":[36],"defined":[37,102],"on":[38,103],"complete":[40],"metric":[41],"space":[42],"$M$":[43,51,67,75,94],"fails":[44],"if":[47,50,66],"and":[48,74],"only":[49],"is":[52],"equivalent":[54],"to":[55,122],"subset":[57],"$\\mathbb{R}$":[59,73],"with":[60],"positive":[61,77],"Lebesgue":[62],"measure,":[63],"or":[64],"equivalently,":[65],"admits":[68],"embedding":[71],"into":[72],"has":[76],"1-dimensional":[78],"Hausdorff":[79],"measure.":[80],"As":[81],"consequence,":[83],"we":[84],"provide":[85],"an":[86],"isometric":[87],"characterization":[88],"pure":[91],"1-unrectifiability":[92],"maps":[101],"copies":[105],"closed":[107],"subsets":[108],"$M$.":[110],"Several":[111],"counterexamples":[112],"showing":[113],"cannot":[118],"be":[119],"naturally":[120],"extended":[121],"vector-valued":[124],"setting":[125],"are":[126],"also":[127],"presented.":[128]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-12T00:00:00"}
