{"id":"https://openalex.org/W2607913557","doi":"https://doi.org/10.3390/e20030185","title":"A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications","display_name":"A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications","publication_year":2018,"publication_date":"2018-03-09","ids":{"openalex":"https://openalex.org/W2607913557","doi":"https://doi.org/10.3390/e20030185","mag":"2607913557","pmid":"https://pubmed.ncbi.nlm.nih.gov/33265276"},"language":"en","primary_location":{"id":"doi:10.3390/e20030185","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20030185","pdf_url":"https://www.mdpi.com/1099-4300/20/3/185/pdf?version=1520849135","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/20/3/185/pdf?version=1520849135","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062208766","display_name":"Arnaud Marsiglietti","orcid":"https://orcid.org/0000-0002-7459-0547"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Arnaud Marsiglietti","raw_affiliation_strings":["Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA 91125, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA 91125, USA","institution_ids":["https://openalex.org/I122411786"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5001633086","display_name":"Victoria Kostina","orcid":"https://orcid.org/0000-0002-2406-7440"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Victoria Kostina","raw_affiliation_strings":["Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA","institution_ids":["https://openalex.org/I122411786"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5062208766"],"corresponding_institution_ids":["https://openalex.org/I122411786"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":5.4116,"has_fulltext":true,"cited_by_count":44,"citation_normalized_percentile":{"value":0.96883764,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"20","issue":"3","first_page":"185","last_page":"185"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10964","display_name":"Wireless Communication Security Techniques","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9794999957084656,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/mathematics","display_name":"Mathematics","score":0.8003641366958618},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6823362708091736},{"id":"https://openalex.org/keywords/differential-entropy","display_name":"Differential entropy","score":0.6451677083969116},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5463748574256897},{"id":"https://openalex.org/keywords/concave-function","display_name":"Concave function","score":0.5417912006378174},{"id":"https://openalex.org/keywords/distortion","display_name":"Distortion (music)","score":0.5329540371894836},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5006082057952881},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.46849972009658813},{"id":"https://openalex.org/keywords/rate\u2013distortion-theory","display_name":"Rate\u2013distortion theory","score":0.4533133804798126},{"id":"https://openalex.org/keywords/entropy-power-inequality","display_name":"Entropy power inequality","score":0.4489063322544098},{"id":"https://openalex.org/keywords/channel-capacity","display_name":"Channel capacity","score":0.44755908846855164},{"id":"https://openalex.org/keywords/multivariate-random-variable","display_name":"Multivariate random variable","score":0.44738438725471497},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4325312674045563},{"id":"https://openalex.org/keywords/second-moment-of-area","display_name":"Second moment of area","score":0.41053518652915955},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3530312180519104},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31111204624176025},{"id":"https://openalex.org/keywords/entropy-rate","display_name":"Entropy rate","score":0.25161537528038025},{"id":"https://openalex.org/keywords/binary-entropy-function","display_name":"Binary entropy function","score":0.2503812909126282},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1950072944164276},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1541862189769745},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.1447860598564148},{"id":"https://openalex.org/keywords/coding","display_name":"Coding (social sciences)","score":0.1401439905166626},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1259538233280182}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8003641366958618},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6823362708091736},{"id":"https://openalex.org/C124551494","wikidata":"https://www.wikidata.org/wiki/Q3055345","display_name":"Differential entropy","level":4,"score":0.6451677083969116},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5463748574256897},{"id":"https://openalex.org/C66690126","wikidata":"https://www.wikidata.org/wiki/Q2914302","display_name":"Concave function","level":3,"score":0.5417912006378174},{"id":"https://openalex.org/C126780896","wikidata":"https://www.wikidata.org/wiki/Q899871","display_name":"Distortion (music)","level":4,"score":0.5329540371894836},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5006082057952881},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.46849972009658813},{"id":"https://openalex.org/C64185310","wikidata":"https://www.wikidata.org/wiki/Q843483","display_name":"Rate\u2013distortion theory","level":3,"score":0.4533133804798126},{"id":"https://openalex.org/C320580","wikidata":"https://www.wikidata.org/wiki/Q5380810","display_name":"Entropy power inequality","level":5,"score":0.4489063322544098},{"id":"https://openalex.org/C97744766","wikidata":"https://www.wikidata.org/wiki/Q870845","display_name":"Channel capacity","level":3,"score":0.44755908846855164},{"id":"https://openalex.org/C138405894","wikidata":"https://www.wikidata.org/wiki/Q3179949","display_name":"Multivariate random variable","level":3,"score":0.44738438725471497},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4325312674045563},{"id":"https://openalex.org/C61062188","wikidata":"https://www.wikidata.org/wiki/Q835065","display_name":"Second moment of area","level":2,"score":0.41053518652915955},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3530312180519104},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31111204624176025},{"id":"https://openalex.org/C125252325","wikidata":"https://www.wikidata.org/wiki/Q1345213","display_name":"Entropy rate","level":4,"score":0.25161537528038025},{"id":"https://openalex.org/C44415725","wikidata":"https://www.wikidata.org/wiki/Q4913893","display_name":"Binary entropy function","level":3,"score":0.2503812909126282},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1950072944164276},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1541862189769745},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.1447860598564148},{"id":"https://openalex.org/C179518139","wikidata":"https://www.wikidata.org/wiki/Q5140297","display_name":"Coding (social sciences)","level":2,"score":0.1401439905166626},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1259538233280182},{"id":"https://openalex.org/C49040817","wikidata":"https://www.wikidata.org/wiki/Q193091","display_name":"Optoelectronics","level":1,"score":0.0},{"id":"https://openalex.org/C46362747","wikidata":"https://www.wikidata.org/wiki/Q173431","display_name":"CMOS","level":2,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C194257627","wikidata":"https://www.wikidata.org/wiki/Q211554","display_name":"Amplifier","level":3,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.3390/e20030185","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20030185","pdf_url":"https://www.mdpi.com/1099-4300/20/3/185/pdf?version=1520849135","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:33265276","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33265276","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:arXiv.org:1704.07766","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1704.07766","pdf_url":"https://arxiv.org/pdf/1704.07766","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:authors.library.caltech.edu:86116","is_oa":true,"landing_page_url":"https://authors.library.caltech.edu/86116/","pdf_url":null,"source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"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":"","raw_type":"Article"},{"id":"pmh:oai:doaj.org/article:13ce1c0b9c4643cc83261a5fe735f29d","is_oa":true,"landing_page_url":"https://doaj.org/article/13ce1c0b9c4643cc83261a5fe735f29d","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 20, Iss 3, p 185 (2018)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/20/3/185/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e20030185","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Entropy; Volume 20; Issue 3; Pages: 185","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7512702","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7512702","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e20030185","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20030185","pdf_url":"https://www.mdpi.com/1099-4300/20/3/185/pdf?version=1520849135","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4486700959","display_name":"CRII: CIF: Data compression with low distortion and finite blocklength","funder_award_id":"1566567","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6909155701","display_name":null,"funder_award_id":"CCF-1566567","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2607913557.pdf","grobid_xml":"https://content.openalex.org/works/W2607913557.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W144802579","https://openalex.org/W1521305480","https://openalex.org/W1556911442","https://openalex.org/W1766348922","https://openalex.org/W1852636839","https://openalex.org/W1966627784","https://openalex.org/W1969388298","https://openalex.org/W1970746657","https://openalex.org/W1995875735","https://openalex.org/W2030458706","https://openalex.org/W2039535673","https://openalex.org/W2043371182","https://openalex.org/W2044246986","https://openalex.org/W2056638576","https://openalex.org/W2071728535","https://openalex.org/W2075360622","https://openalex.org/W2084071523","https://openalex.org/W2087867172","https://openalex.org/W2099111195","https://openalex.org/W2121952483","https://openalex.org/W2131908288","https://openalex.org/W2133318567","https://openalex.org/W2134887275","https://openalex.org/W2144094762","https://openalex.org/W2146519989","https://openalex.org/W2157962569","https://openalex.org/W2169690298","https://openalex.org/W2201108895","https://openalex.org/W2308818299","https://openalex.org/W2312825124","https://openalex.org/W2336140998","https://openalex.org/W2488280448","https://openalex.org/W2530473264","https://openalex.org/W2542124682","https://openalex.org/W2592082291","https://openalex.org/W2742234152","https://openalex.org/W2962982621","https://openalex.org/W2963952097","https://openalex.org/W2964340705","https://openalex.org/W2993383518","https://openalex.org/W3098937466","https://openalex.org/W3105202125","https://openalex.org/W4293438359","https://openalex.org/W4300859130","https://openalex.org/W6640802941"],"related_works":["https://openalex.org/W2912008365","https://openalex.org/W1968021238","https://openalex.org/W2963235040","https://openalex.org/W2975007897","https://openalex.org/W2107271148","https://openalex.org/W2034723588","https://openalex.org/W2136464164","https://openalex.org/W2520215844","https://openalex.org/W4301387713","https://openalex.org/W2269447256"],"abstract_inverted_index":{"We":[0,140,157],"derive":[1],"a":[2,10,29,199],"lower":[3,96],"bound":[4,26,97],"on":[5,42,144],"the":[6,18,43,47,53,87,90,94,115,123,145,153,160,163,169,172,176,196],"differential":[7],"entropy":[8,31],"of":[9,17,22,112,138,147,165,171,198],"log-concave":[11,57],"random":[12,200],"variable":[13],"X":[14,202],"in":[15],"terms":[16],"p-th":[19],"absolute":[20],"moment":[21],"X.":[23],"The":[24],"new":[25,40],"leads":[27],"to":[28,39,195],"reverse":[30],"power":[32,179],"inequality":[33],"with":[34,78,175,203],"an":[35],"explicit":[36],"constant,":[37],"and":[38,46,59,83,93,114,168],"bounds":[41,143],"rate-distortion":[44,54,91],"function":[45,55,92],"channel":[48,174],"capacity.":[49],"Specifically,":[50],"we":[51,84],"study":[52],"for":[56],"sources":[58],"distortion":[60,117],"measure":[61],"d":[62],"(":[63,102,129,184],"x":[64,66,71,73],",":[65,77,82],"^":[67,74],")":[68,105,133,188],"=":[69],"|":[70,75],"\u2212":[72],"r":[76,79,113],"\u2265":[80],"1":[81,107,135,190],"establish":[85],"that":[86,159],"difference":[88,124,161],"between":[89,162],"Shannon":[95],"is":[98,125,155,180],"at":[99,126,181],"most":[100,127,182],"log":[101,128,183],"\u03c0":[103,130,185],"e":[104,131,186],"\u2248":[106,134,189],".":[108],"5":[109],"bits,":[110],"independently":[111],"target":[116],"d.":[118,139],"For":[119],"mean-square":[120],"error":[121],"distortion,":[122],"2":[132,187],"bit,":[136],"regardless":[137],"also":[141],"provide":[142],"capacity":[146,164,170],"memoryless":[148],"additive":[149],"noise":[150,154,178],"channels":[151,167],"when":[152],"log-concave.":[156],"show":[158],"such":[166],"Gaussian":[173],"same":[177],"bit.":[191],"Our":[192,207],"results":[193],"generalize":[194],"case":[197],"vector":[201],"possibly":[204],"dependent":[205],"coordinates.":[206],"proof":[208],"technique":[209],"leverages":[210],"tools":[211],"from":[212],"convex":[213],"geometry.":[214]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":11},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}