{"id":"https://openalex.org/W7203687994","doi":"https://doi.org/10.48550/arxiv.2608.14909","title":"On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube","display_name":"On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube","publication_year":2026,"publication_date":"2026-08-14","ids":{"openalex":"https://openalex.org/W7203687994","doi":"https://doi.org/10.48550/arxiv.2608.14909"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.14909","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.14909","pdf_url":"https://arxiv.org/pdf/2608.14909","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2608.14909","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020875207","display_name":"Gennadiy Averkov","orcid":"https://orcid.org/0000-0003-2245-9958","observed_orcids":["https://orcid.org/0000-0003-2245-9958"]},"institutions":[{"id":"https://openalex.org/I51783024","display_name":"Brandenburg University of Technology Cottbus-Senftenberg","ror":"https://ror.org/02wxx3e24","country_code":"DE","type":"education","lineage":["https://openalex.org/I51783024"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Averkov, Gennadiy","raw_affiliation_strings":["Fakult\u00e4t 1 , BTU Cottbus-Senftenberg , Platz der Deutschen Einheit 1 , 03046 Cottbus , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t 1 , BTU Cottbus-Senftenberg , Platz der Deutschen Einheit 1 , 03046 Cottbus , Germany","institution_ids":["https://openalex.org/I51783024"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068864415","display_name":"Katherina von Dichter","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I102607778","display_name":"Cleveland State University","ror":"https://ror.org/002tx1f22","country_code":"US","type":"education","lineage":["https://openalex.org/I102607778"]},{"id":"https://openalex.org/I51783024","display_name":"Brandenburg University of Technology Cottbus-Senftenberg","ror":"https://ror.org/02wxx3e24","country_code":"DE","type":"education","lineage":["https://openalex.org/I51783024"]}],"countries":["DE","US"],"is_corresponding":false,"raw_author_name":"von Dichter, Katherina","raw_affiliation_strings":["Department of Mathematics and Statistics , Cleveland State University , 2121 Euclid Ave , Cleveland , Ohio , 44115 USA","Fakult\u00e4t 1 , BTU Cottbus-Senftenberg , Platz der Deutschen Einheit 1 , 03046 Cottbus , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , Cleveland State University , 2121 Euclid Ave , Cleveland , Ohio , 44115 USA","institution_ids":["https://openalex.org/I102607778"]},{"raw_affiliation_string":"Fakult\u00e4t 1 , BTU Cottbus-Senftenberg , Platz der Deutschen Einheit 1 , 03046 Cottbus , Germany","institution_ids":["https://openalex.org/I51783024"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059943366","display_name":"Ivan Soprunov","orcid":"https://orcid.org/0000-0003-0122-7699","observed_orcids":["https://orcid.org/0000-0003-0122-7699"]},"institutions":[{"id":"https://openalex.org/I51783024","display_name":"Brandenburg University of Technology Cottbus-Senftenberg","ror":"https://ror.org/02wxx3e24","country_code":"DE","type":"education","lineage":["https://openalex.org/I51783024"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Soprunov, Ivan","raw_affiliation_strings":["Fakult\u00e4t 1 , BTU Cottbus-Senftenberg , Platz der Deutschen Einheit 1 , 03046 Cottbus , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t 1 , BTU Cottbus-Senftenberg , Platz der Deutschen Einheit 1 , 03046 Cottbus , Germany","institution_ids":["https://openalex.org/I51783024"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T11830","display_name":"Point processes and geometric inequalities","score":0.9980000257492065,"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"}},"topics":[{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9980000257492065,"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.0013000000035390258,"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":0.0006000000284984708,"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/zonoids","display_name":"zonoids","score":0.9940000176429749},{"id":"https://openalex.org/keywords/loomis-whitney-inequality","display_name":"Loomis-Whitney inequality","score":0.949999988079071},{"id":"https://openalex.org/keywords/copositivity","display_name":"copositivity","score":0.9769999980926514},{"id":"https://openalex.org/keywords/sum-of-squares-certificates","display_name":"sum-of-squares certificates","score":0.9049999713897705},{"id":"https://openalex.org/keywords/matrix-inequalities","display_name":"matrix inequalities","score":0.5699999928474426},{"id":"https://openalex.org/keywords/convex-bodies","display_name":"convex bodies","score":0.3880000114440918},{"id":"https://openalex.org/keywords/information-theory","display_name":"information theory","score":0.3840000033378601}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8245999813079834},{"id":"https://openalex.org/C50820777","wikidata":"https://www.wikidata.org/wiki/Q213723","display_name":"Hypercube","level":2,"score":0.6930999755859375},{"id":"https://openalex.org/C106286213","wikidata":"https://www.wikidata.org/wiki/Q898572","display_name":"Matroid","level":2,"score":0.6643999814987183},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.620199978351593},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5734000205993652},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5273000001907349},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4828000068664551},{"id":"https://openalex.org/C126087398","wikidata":"https://www.wikidata.org/wiki/Q1149189","display_name":"Rearrangement inequality","level":4,"score":0.476500004529953},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.46619999408721924},{"id":"https://openalex.org/C46979015","wikidata":"https://www.wikidata.org/wiki/Q6666859","display_name":"Log sum inequality","level":3,"score":0.42289999127388},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3978999853134155},{"id":"https://openalex.org/C84135550","wikidata":"https://www.wikidata.org/wiki/Q3154022","display_name":"Linear inequality","level":3,"score":0.39640000462532043},{"id":"https://openalex.org/C206399364","wikidata":"https://www.wikidata.org/wiki/Q3154001","display_name":"Kantorovich inequality","level":4,"score":0.36719998717308044},{"id":"https://openalex.org/C44550424","wikidata":"https://www.wikidata.org/wiki/Q4424496","display_name":"Mixed volume","level":5,"score":0.36309999227523804},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.3540000021457672},{"id":"https://openalex.org/C91558832","wikidata":"https://www.wikidata.org/wiki/Q731894","display_name":"H\u00f6lder's inequality","level":4,"score":0.3343000113964081},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2948000133037567},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.29010000824928284},{"id":"https://openalex.org/C102379996","wikidata":"https://www.wikidata.org/wiki/Q755092","display_name":"Minkowski inequality","level":5,"score":0.27309998869895935},{"id":"https://openalex.org/C188791865","wikidata":"https://www.wikidata.org/wiki/Q366730","display_name":"Ky Fan inequality","level":5,"score":0.2712000012397766},{"id":"https://openalex.org/C184701930","wikidata":"https://www.wikidata.org/wiki/Q190546","display_name":"Cauchy\u2013Schwarz inequality","level":3,"score":0.26460000872612},{"id":"https://openalex.org/C177180624","wikidata":"https://www.wikidata.org/wiki/Q728662","display_name":"Bernoulli's inequality","level":5,"score":0.2547000050544739},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.25099998712539673}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.14909","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.14909","pdf_url":"https://arxiv.org/pdf/2608.14909","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.14909","doi":"https://doi.org/10.48550/arxiv.2608.14909","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.14909","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.14909","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.14909","pdf_url":"https://arxiv.org/pdf/2608.14909","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7203687994.pdf","grobid_xml":"https://content.openalex.org/works/W7203687994.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W4327693987","https://openalex.org/W4416735854","https://openalex.org/W4417009289","https://openalex.org/W7080306982","https://openalex.org/W7164924697","https://openalex.org/W7202178085","https://openalex.org/W7203454531"],"related_works":["https://openalex.org/W7202178085","https://openalex.org/W7203454531","https://openalex.org/W4388686790","https://openalex.org/W2008565479","https://openalex.org/W4403799661","https://openalex.org/W4384111101","https://openalex.org/W4410627445","https://openalex.org/W2043087417","https://openalex.org/W1978249312","https://openalex.org/W2320022382"],"abstract_inverted_index":{"In":[0,89,120,163],"convex":[1,14,67],"geometry,":[2],"a":[3,17,30,38,71,81,136,160,166,194],"number":[4],"of":[5,10,13,26,37,46,56,84,135,152,168,170,199],"inequalities":[6,25,55,181],"for":[7,61,65,87,100],"the":[8,35,50,97,142,150,153,171],"volume":[9],"coordinate":[11],"projections":[12],"bodies":[15],"share":[16],"common":[18],"structural":[19],"property.":[20],"We":[21,79,129,192],"formalize":[22],"them":[23],"as":[24],"log-submodularity":[27,57,85,98],"type,":[28],"providing":[29,159],"definition":[31],"that":[32,174],"stems":[33],"from":[34],"notion":[36],"submodular":[39],"function.":[40],"The":[41],"Loomis-Whitney":[42],"inequality":[43,83,93,123,132],"and":[44,76,107,115,178,187,196],"some":[45],"its":[47],"generalizations":[48],"are":[49,59],"most":[51],"prominent":[52],"examples.":[53],"Some":[54],"type":[58,86],"valid":[60],"zonoids":[62,101],"but":[63],"not":[64],"general":[66],"bodies.":[68],"They":[69],"have":[70],"natural":[72],"meaning":[73],"in":[74,112,117,126],"statistics":[75],"information":[77],"theory.":[78],"propose":[80],"new":[82],"zonoids.":[88],"dimension":[90,113],"3,":[91],"our":[92,122,131,190],"is":[94,110,124],"equivalent":[95],"to":[96,133,189],"conjecture":[99],"formulated":[102],"by":[103,158,185],"Fradelizi,":[104],"Madiman,":[105],"Meyer,":[106],"Zvavitch,":[108],"which":[109,139],"true":[111,125],"$3$":[114],"false":[116],"higher":[118],"dimensions.":[119],"contrast,":[121],"every":[127,164],"dimension.":[128],"reduce":[130],"copositivity":[134,151,169],"special":[137],"polynomial,":[138],"we":[140,148],"call":[141],"hypercube":[143,155,172],"polynomial.":[144],"For":[145],"4-dimensional":[146],"zonoids,":[147],"verify":[149],"respective":[154],"polynomial":[156,173],"algebraically":[157],"sum-of-squares":[161],"certificate.":[162],"dimension,":[165],"proof":[167],"uses":[175],"probability":[176],"theory":[177],"well-known":[179],"matrix":[180],"has":[182],"been":[183],"produced":[184],"LLMs":[186],"brought":[188],"attention.":[191],"present":[193],"detailed":[195],"easy-to-read":[197],"version":[198],"this":[200],"proof.":[201]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-08-19T00:00:00"}
