{"id":"https://openalex.org/W7213249197","doi":"https://doi.org/10.48550/arxiv.2609.15817","title":"On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson","display_name":"On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson","publication_year":2026,"publication_date":"2026-09-14","ids":{"openalex":"https://openalex.org/W7213249197","doi":"https://doi.org/10.48550/arxiv.2609.15817"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.15817","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.15817","pdf_url":"https://arxiv.org/pdf/2609.15817","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/2609.15817","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151796357","display_name":"Lixin Cheng","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cheng, Lixin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5063236639","display_name":"Qingjin Cheng","orcid":"https://orcid.org/0000-0001-8015-877X","observed_orcids":["https://orcid.org/0000-0001-8015-877X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cheng, Qingjin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5151797801","display_name":"Yue Wang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Yue","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":0.9990000128746033,"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":0.9990000128746033,"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":0.00019999999494757503,"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":9.999999747378752e-05,"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/c0","display_name":"c0","score":0.8180000185966492},{"id":"https://openalex.org/keywords/unit-sphere","display_name":"unit sphere","score":0.5370000004768372},{"id":"https://openalex.org/keywords/unit-ball","display_name":"unit ball","score":0.6679999828338623},{"id":"https://openalex.org/keywords/lipschitz-functions","display_name":"Lipschitz functions","score":0.6959999799728394},{"id":"https://openalex.org/keywords/isomorphism","display_name":"isomorphism","score":0.43700000643730164},{"id":"https://openalex.org/keywords/banach-spaces","display_name":"Banach spaces","score":0.5529999732971191}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7868000268936157},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.769599974155426},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.7135999798774719},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.6710000038146973},{"id":"https://openalex.org/C191948623","wikidata":"https://www.wikidata.org/wiki/Q1307800","display_name":"Unit sphere","level":2,"score":0.6032000184059143},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5282999873161316},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.45660001039505005},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.45179998874664307},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.4122999906539917},{"id":"https://openalex.org/C152306953","wikidata":"https://www.wikidata.org/wiki/Q621744","display_name":"Approximation property","level":3,"score":0.3984000086784363},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36660000681877136},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.3564999997615814},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.28679999709129333},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2721000015735626},{"id":"https://openalex.org/C84316537","wikidata":"https://www.wikidata.org/wiki/Q36255","display_name":"Unit vector","level":2,"score":0.2660999894142151}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.15817","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.15817","pdf_url":"https://arxiv.org/pdf/2609.15817","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.2609.15817","doi":"https://doi.org/10.48550/arxiv.2609.15817","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.15817","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:2609.15817","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.15817","pdf_url":"https://arxiv.org/pdf/2609.15817","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/G4146126967","display_name":null,"funder_award_id":"12271453","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8864014117","display_name":null,"funder_award_id":"12071389","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213249197.pdf","grobid_xml":"https://content.openalex.org/works/W7213249197.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W1969564788","https://openalex.org/W2110588906","https://openalex.org/W2345472561","https://openalex.org/W2746150456","https://openalex.org/W7169789285"],"related_works":["https://openalex.org/W2746150456","https://openalex.org/W2563707238","https://openalex.org/W2037678166","https://openalex.org/W2519137546","https://openalex.org/W1970605944","https://openalex.org/W7220741616","https://openalex.org/W2050198994","https://openalex.org/W2950151614","https://openalex.org/W2056870732","https://openalex.org/W1547860832"],"abstract_inverted_index":{"Let":[0,128],"$X$":[1,17],"be":[2,8],"a":[3,48,161],"normed":[4],"space":[5,73],"and":[6,29,45,130,145],"$S_X$":[7],"the":[9,56,71,104,112,133,136,172,178],"unit":[10,183],"sphere":[11],"of":[12,33,75,111,115,135,139,163,181],"$X$.":[13],"We":[14,101],"say":[15],"that":[16,38,55,91],"has":[18],"Property":[19,80,84,95,99],"(H)":[20,96],"if":[21],"there":[22,46],"exist":[23],"two":[24],"increasing":[25],"sequences":[26,78],"$V_j\\subset":[27],"X$":[28],"$H_j\\subset":[30],"\\ell_2$,":[31],"$j=1,2,\\ldots$,":[32,59],"finite":[34],"dimensional":[35],"subspaces":[36],"such":[37,54],"$V=\\bigcup_j":[39],"V_j$":[40],"is":[41,47],"dense":[42],"in":[43],"$X$,":[44],"uniformly":[49],"continuous":[50],"mapping":[51],"$\u03c8:S_V\\to":[52],"S_{\\ell_2}$":[53],"restrictions":[57],"$\u03c8|_{S_{V_j}}$,":[58],"are":[60],"isomorphisms":[61],"from":[62,119,141,146],"$S_{V_j}$":[63],"onto":[64,143,148],"$S_{H_j}$.":[65],"The":[66,108,157],"Kasparov-Yu":[67],"problem":[68,162,176],"asks":[69],"whether":[70],"Banach":[72],"$c_0$":[74,92],"all":[76,116],"null":[77],"admits":[79,93],"(H),":[81],"or":[82],"rational":[83,98],"(H).":[85,100],"In":[86],"this":[87],"paper,":[88],"we":[89],"prove":[90],"neither":[94],"nor":[97],"also":[102],"obtain":[103],"following":[105],"results.":[106],"(1)":[107],"infimum":[109],"$\u03bc_n$":[110],"Lipschitz":[113,137,179],"constants":[114,138],"nonzero-degree":[117,166],"maps":[118,167],"$S_{\\ell_\\infty^n}$":[120,142],"to":[121],"$S_{\\ell_2^n}$":[122,144],"satisfies":[123],"$\\lim_{n\\to\\infty}":[124,152],"\u03bc_n/\\log":[125],"n=1/2$.":[126,156],"(2)":[127],"$\u03bb_n$":[129],"$C_n$":[131],"denote":[132],"infima":[134],"homeomorphisms":[140],"$B_{\\ell_\\infty^n}$":[147],"$B_{\\ell_2^n}$,":[149],"respectively.":[150],"Then":[151],"\u03bb_n/\\log":[153],"n=\\lim_{n\\to\\infty}":[154],"C_n/\\log":[155],"first":[158],"result":[159],"solves":[160],"Gromov":[164],"on":[165,177],"between":[168],"finite-dimensional":[169,182],"spheres,":[170],"while":[171],"second":[173],"settles":[174],"Johnson's":[175],"distortion":[180],"balls.":[184]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-16T00:00:00"}