{"id":"https://openalex.org/W7225318409","doi":"https://doi.org/10.5281/zenodo.23969381","title":"Cubic Median Inequalities and the Power Conjecture of Liu (2024)","display_name":"Cubic Median Inequalities and the Power Conjecture of Liu (2024)","publication_year":2026,"publication_date":"2026-11-05","ids":{"openalex":"https://openalex.org/W7225318409","doi":"https://doi.org/10.5281/zenodo.23969381"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23969381","doi":"https://doi.org/10.5281/zenodo.23969381","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23969381","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23969381","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5094149092","display_name":"Christos Panorias","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Panorias, Christos","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5094149092"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T12504","display_name":"Mathematics and Applications","score":0.5673999786376953,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12504","display_name":"Mathematics and Applications","score":0.5673999786376953,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.4235999882221222,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0012000000569969416,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/triangle-medians","display_name":"triangle medians","score":0.8859999775886536},{"id":"https://openalex.org/keywords/power-mean-inequality","display_name":"power mean inequality","score":0.5120000243186951},{"id":"https://openalex.org/keywords/acute-triangles","display_name":"acute triangles","score":0.6420000195503235},{"id":"https://openalex.org/keywords/sharp-constants","display_name":"sharp constants","score":0.6460000276565552},{"id":"https://openalex.org/keywords/computer-assisted-proof","display_name":"computer-assisted proof","score":0.6830000281333923},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.6349999904632568},{"id":"https://openalex.org/keywords/geometric-inequalities","display_name":"geometric inequalities","score":0.4869999885559082},{"id":"https://openalex.org/keywords/triangle-geometry","display_name":"triangle geometry","score":0.3409999907016754},{"id":"https://openalex.org/keywords/ai-assisted-code-generation","display_name":"AI-assisted code generation","score":0.5260000228881836},{"id":"https://openalex.org/keywords/ai-assisted-data-analysis","display_name":"AI-assisted data analysis","score":0.46700000762939453},{"id":"https://openalex.org/keywords/ai-assisted-scientific-writing","display_name":"AI-assisted scientific writing","score":0.515999972820282}],"concepts":[{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7985000014305115},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7953000068664551},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.6626999974250793},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.640999972820282},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.5928999781608582},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4984999895095825},{"id":"https://openalex.org/C43169469","wikidata":"https://www.wikidata.org/wiki/Q580922","display_name":"Preprint","level":2,"score":0.4747999906539917},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4440999925136566},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.42160001397132874},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.4077000021934509},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.4034000039100647},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.3991999924182892},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3882000148296356},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.3880999982357025},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.3864000141620636},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.38370001316070557},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.3310999870300293},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.31839999556541443},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.27250000834465027},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2648000121116638},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.25429999828338623},{"id":"https://openalex.org/C160145156","wikidata":"https://www.wikidata.org/wiki/Q778586","display_name":"Executable","level":2,"score":0.2531999945640564}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.5281/zenodo.23969381","doi":"https://doi.org/10.5281/zenodo.23969381","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23969381","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23969382","doi":"https://doi.org/10.5281/zenodo.23969382","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23969382","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23969758","doi":"https://doi.org/10.5281/zenodo.23969758","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23969758","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.23969381","doi":"https://doi.org/10.5281/zenodo.23969381","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23969381","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2183999119","https://openalex.org/W7225333086","https://openalex.org/W7223159669","https://openalex.org/W7225117500","https://openalex.org/W7223118351","https://openalex.org/W4409894694","https://openalex.org/W7099543666","https://openalex.org/W200452182","https://openalex.org/W4388543365","https://openalex.org/W2328088311"],"abstract_inverted_index":{"This":[0,128],"consolidated":[1],"preprint":[2,71,196],"proves":[3],"Liu's":[4],"cubic":[5],"median":[6,63],"inequality":[7],"with":[8,22],"optimal":[9],"coefficient":[10,48],"1/72":[11],"and":[12,26,50,61,81,90,96,116,134,160,181,187,192,205],"classifies":[13],"the":[14,55,106,109,131,138,145,177,189],"universal":[15],"positive-exponent":[16],"set":[17],"as":[18],"a":[19],"closed":[20],"interval":[21,52],"endpoints":[23],"approximately":[24],"0.0589422833021817898":[25],"4.6683652379684839567.":[27],"The":[28,35,58,78,93],"positive":[29],"branch":[30,38],"concerns":[31,130],"all":[32],"nondegenerate":[33,41],"triangles.":[34,43],"reversed":[36],"negative-exponent":[37],"holds":[39],"on":[40,105],"nonobtuse":[42],"Equality":[44],"cases,":[45],"counterexamples,":[46],"exact":[47],"certificates":[49,53],"outward":[51],"accompany":[54],"computer-assisted":[56],"proofs.":[57],"original":[59,110],"conjectures":[60],"earlier":[62],"inequalities":[64],"are":[65,103,208],"credited":[66],"to":[67],"Jian":[68],"Liu.":[69],"Consolidated":[70],"edition":[72],"1.0,":[73],"dated":[74],"5":[75],"November":[76],"2026.":[77],"main":[79],"PDF":[80],"its":[82],"source/certificate":[83],"archive":[84],"form":[85],"one":[86],"article.":[87],"Conceptual":[88],"origin":[89,133],"AI":[91],"assistance.":[92],"initial":[94],"idea":[95],"conceptual":[97,132],"starting":[98],"point":[99],"of":[100,108,122,137,156],"this":[101],"investigation":[102],"based":[104],"framework":[107],"geometric":[111],"paper":[112],"by":[113,144],"Christos":[114,174],"Panorias":[115,175],"Kostandina":[117],"Theodhori,":[118],"Canonical":[119],"Parametric":[120],"Representation":[121],"Plane":[123],"Euclidean":[124],"Geometry":[125],"(DOI:":[126],"10.5281/zenodo.23837735).":[127],"attribution":[129],"methodological":[135],"organization":[136],"research;":[139],"every":[140],"mathematical":[141,169,190],"premise":[142],"required":[143],"present":[146],"proofs":[147],"is":[148,176],"stated":[149],"explicitly":[150],"here.":[151],"OpenAI":[152],"ChatGPT/Codex":[153],"assisted":[154],"verification":[155],"calculations,":[157],"editorial":[158],"polishing":[159],"selected,":[161],"delimited":[162],"arguments":[163],"during":[164],"proof":[165],"development,":[166],"including":[167],"substantive":[168],"reasoning":[170],"in":[171],"those":[172],"arguments.":[173],"sole":[178],"named":[179],"author":[180],"retains":[182],"responsibility":[183],"for":[184,197],"reviewing,":[185],"adopting":[186],"submitting":[188],"claims":[191],"final":[193],"text.":[194],"Research":[195],"scholarly":[198],"scrutiny.":[199],"Independent":[200],"peer":[201],"review,":[202],"proof-assistant":[203],"formalization":[204],"worldwide":[206],"priority":[207],"not":[209],"certified.":[210]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-06T00:00:00"}