{"id":"https://openalex.org/W7129303548","doi":"https://doi.org/10.48550/arxiv.2602.14852","title":"Lower Estimates for $L_1$-Distortion of Transportation Cost Spaces","display_name":"Lower Estimates for $L_1$-Distortion of Transportation Cost Spaces","publication_year":2026,"publication_date":"2026-02-16","ids":{"openalex":"https://openalex.org/W7129303548","doi":"https://doi.org/10.48550/arxiv.2602.14852"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2602.14852","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112097895","display_name":"Chris Gartland","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gartland, Chris","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5024496853","display_name":"Mikhail I. Ostrovskii","orcid":"https://orcid.org/0000-0002-7164-196X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ostrovskii, Mikhail","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.2558000087738037,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.2558000087738037,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.10090000182390213,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.0754999965429306,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6234999895095825},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.5587999820709229},{"id":"https://openalex.org/keywords/matching","display_name":"Matching (statistics)","score":0.5315999984741211},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.5157999992370605},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5098999738693237},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.5077999830245972},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.4887000024318695},{"id":"https://openalex.org/keywords/triangle-inequality","display_name":"Triangle inequality","score":0.45170000195503235},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.4390999972820282}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7364000082015991},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6234999895095825},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.5587999820709229},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.5315999984741211},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5157999992370605},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5098999738693237},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5077999830245972},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.4887000024318695},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.45170000195503235},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.4390999972820282},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.4316999912261963},{"id":"https://openalex.org/C2779494224","wikidata":"https://www.wikidata.org/wiki/Q5645799","display_name":"Hamming space","level":5,"score":0.40630000829696655},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39579999446868896},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37310001254081726},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.35899999737739563},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3517000079154968},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.35089999437332153},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.34200000762939453},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.3384999930858612},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3093999922275543},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.29120001196861267},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.2775000035762787},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.2720000147819519},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.2696000039577484},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2623000144958496},{"id":"https://openalex.org/C197947376","wikidata":"https://www.wikidata.org/wiki/Q5155608","display_name":"Comparability","level":2,"score":0.25619998574256897},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.25429999828338623}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2602.14852","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2602.14852","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.14852","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:doi:10.48550/arxiv.2602.14852","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Quantifying":[0],"the":[1,33,61,73,96,108,113,118,121,144,157,163,170,183,200,203,207,215,242,264],"degree":[2],"of":[3,58,76,85,98,116,120,123,129,146,172,185,202,254,268],"dissimilarity":[4,26],"between":[5],"two":[6,140],"probability":[7],"distributions":[8,125],"on":[9,29,126,151,206,241],"a":[10,15,67,187,234],"finite":[11],"metric":[12,86],"space":[13,63,122],"is":[14,32,193,199,233,247],"fundamental":[16],"task":[17],"in":[18,136,271],"Computer":[19,22,99],"Science":[20],"and":[21,49,51,148,168,174,259],"Vision.":[23],"A":[24,38],"natural":[25],"measure":[27],"based":[28],"optimal":[30],"transport":[31],"Earth":[34],"Mover's":[35],"Distance":[36],"(EMD).":[37],"key":[39,68,141,224],"technique":[40],"for":[41,82,159,177,186,209,238],"analyzing":[42],"this":[43],"metric,":[44],"pioneered":[45],"by":[46,190],"Charikar":[47],"(2002)":[48],"Indyk":[50,111],"Thaper":[52],"(2003),":[53],"involves":[54],"constructing":[55],"low-distortion":[56,91],"embeddings":[57,92],"EMD(X)":[59],"into":[60,93],"Lebesgue":[62],"$L_1$.":[64,94],"It":[65],"became":[66],"problem":[69,115],"to":[70,89,211,220,228,250],"investigate":[71],"whether":[72],"upper":[74,166,217],"bound":[75,205,218],"$O(\\log":[77],"n)$":[78],"can":[79],"be":[80],"improved":[81],"important":[83],"classes":[84],"spaces":[87],"known":[88],"admit":[90],"In":[95],"context":[97],"Vision,":[100],"grid":[101,192,243],"graphs,":[102,255],"especially":[103],"planar":[104,178,188],"grids,":[105,179],"are":[106],"among":[107],"most":[109],"fundamental.":[110],"posed":[112],"related":[114],"estimating":[117],"$L_1$-distortion":[119,158,184,208],"uniform":[124],"$n$-point":[127],"subsets":[128],"$R^2$.":[130],"The":[131,223],"Progress":[132],"Report,":[133],"last":[134],"updated":[135],"August":[137],"2011,":[138],"highlighted":[139],"results:":[142],"first,":[143],"work":[145],"Khot":[147],"Naor":[149,173],"(2006)":[150],"Hamming":[152,160],"cubes,":[153],"which":[154,180],"showed":[155],"that":[156,182],"cubes":[161],"meets":[162],"described":[164],"above":[165],"estimate,":[167],"second,":[169],"result":[171,198],"Schechtman":[175],"(2007)":[176],"established":[181],"$n$":[189,191],"$\u03a9(\\sqrt{\\log":[194],"n})$.":[195],"Our":[196,245],"first":[197],"improvement":[201],"lower":[204],"grids":[210],"$\u03a9(\\log":[212],"n)$,":[213],"matching":[214],"universal":[216],"up":[219],"multiplicative":[221],"constants.":[222],"ingredient":[225],"allowing":[226],"us":[227],"obtain":[229,263],"these":[230,272],"sharp":[231,265],"estimates":[232,267],"new":[235],"Sobolev-type":[236],"inequality":[237],"scalar-valued":[239],"functions":[240],"graphs.":[244,261],"method":[246],"also":[248],"applicable":[249],"many":[251],"recursive":[252],"families":[253],"such":[256],"as":[257,274],"diamond":[258],"Laakso":[260],"We":[262],"distortion":[266],"$\\log":[269],"n$":[270],"cases":[273],"well.":[275]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-02-18T00:00:00"}
