{"id":"https://openalex.org/W4366774999","doi":"https://doi.org/10.3934/nhm.2023053","title":"Isoperimetric planar clusters with infinitely many regions","display_name":"Isoperimetric planar clusters with infinitely many regions","publication_year":2023,"publication_date":"2023-01-01","ids":{"openalex":"https://openalex.org/W4366774999","doi":"https://doi.org/10.3934/nhm.2023053"},"language":"en","primary_location":{"id":"doi:10.3934/nhm.2023053","is_oa":true,"landing_page_url":"https://doi.org/10.3934/nhm.2023053","pdf_url":null,"source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks and Heterogeneous Media","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.3934/nhm.2023053","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5036258696","display_name":"Matteo Novaga","orcid":"https://orcid.org/0000-0002-0812-3780"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Matteo Novaga","raw_affiliation_strings":["Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy","institution_ids":["https://openalex.org/I108290504"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045145378","display_name":"Emanuele Paolini","orcid":"https://orcid.org/0000-0001-7021-7537"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Emanuele Paolini","raw_affiliation_strings":["Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy","institution_ids":["https://openalex.org/I108290504"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066206471","display_name":"Eugene Stepanov","orcid":"https://orcid.org/0000-0002-0517-155X"},"institutions":[{"id":"https://openalex.org/I1313323035","display_name":"Russian Academy of Sciences","ror":"https://ror.org/05qrfxd25","country_code":"RU","type":"funder","lineage":["https://openalex.org/I1313323035"]},{"id":"https://openalex.org/I118501908","display_name":"National Research University Higher School of Economics","ror":"https://ror.org/055f7t516","country_code":"RU","type":"education","lineage":["https://openalex.org/I118501908"]},{"id":"https://openalex.org/I2801160905","display_name":"Steklov Mathematical Institute","ror":"https://ror.org/03zeg8w71","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I2801160905","https://openalex.org/I4210124601"]},{"id":"https://openalex.org/I157210198","display_name":"Scuola Normale Superiore","ror":"https://ror.org/03aydme10","country_code":"IT","type":"education","lineage":["https://openalex.org/I157210198"]}],"countries":["IT","RU"],"is_corresponding":false,"raw_author_name":"Eugene Stepanov","raw_affiliation_strings":["Faculty of Mathematics, Higher School of Economics, Moscow, Russia","Scuola Normale Superiore, Piazza dei Cavalieri 6, Pisa, Italy","St. Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, St. Petersburg, Russia"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, Higher School of Economics, Moscow, Russia","institution_ids":["https://openalex.org/I118501908"]},{"raw_affiliation_string":"Scuola Normale Superiore, Piazza dei Cavalieri 6, Pisa, Italy","institution_ids":["https://openalex.org/I157210198"]},{"raw_affiliation_string":"St. Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, St. Petersburg, Russia","institution_ids":["https://openalex.org/I2801160905","https://openalex.org/I1313323035"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5036936458","display_name":"Vincenzo Maria Tortorelli","orcid":null},"institutions":[{"id":"https://openalex.org/I2801160905","display_name":"Steklov Mathematical Institute","ror":"https://ror.org/03zeg8w71","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I2801160905","https://openalex.org/I4210124601"]},{"id":"https://openalex.org/I1313323035","display_name":"Russian Academy of Sciences","ror":"https://ror.org/05qrfxd25","country_code":"RU","type":"funder","lineage":["https://openalex.org/I1313323035"]},{"id":"https://openalex.org/I118501908","display_name":"National Research University Higher School of Economics","ror":"https://ror.org/055f7t516","country_code":"RU","type":"education","lineage":["https://openalex.org/I118501908"]},{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]},{"id":"https://openalex.org/I157210198","display_name":"Scuola Normale Superiore","ror":"https://ror.org/03aydme10","country_code":"IT","type":"education","lineage":["https://openalex.org/I157210198"]}],"countries":["IT","RU"],"is_corresponding":true,"raw_author_name":"Vincenzo Maria Tortorelli","raw_affiliation_strings":["Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy","St. Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, St. Petersburg, Russia","Scuola Normale Superiore, Piazza dei Cavalieri 6, Pisa, Italy","Faculty of Mathematics, Higher School of Economics, Moscow, Russia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy","institution_ids":["https://openalex.org/I108290504"]},{"raw_affiliation_string":"St. Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, St. Petersburg, Russia","institution_ids":["https://openalex.org/I2801160905","https://openalex.org/I1313323035"]},{"raw_affiliation_string":"Scuola Normale Superiore, Piazza dei Cavalieri 6, Pisa, Italy","institution_ids":["https://openalex.org/I157210198"]},{"raw_affiliation_string":"Faculty of Mathematics, Higher School of Economics, Moscow, Russia","institution_ids":["https://openalex.org/I118501908"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5036936458"],"corresponding_institution_ids":["https://openalex.org/I108290504","https://openalex.org/I118501908","https://openalex.org/I1313323035","https://openalex.org/I157210198","https://openalex.org/I2801160905"],"apc_list":null,"apc_paid":null,"fwci":3.2197,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.9144966,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"18","issue":"3","first_page":"1226","last_page":"1235"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9991999864578247,"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":0.9991999864578247,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9990000128746033,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9983000159263611,"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/isoperimetric-inequality","display_name":"Isoperimetric inequality","score":0.9317944049835205},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7890989780426025},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.7314302921295166},{"id":"https://openalex.org/keywords/cluster","display_name":"Cluster (spacecraft)","score":0.6335017085075378},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.6145790219306946},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5988266468048096},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.571588397026062},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.502526044845581},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.4541163444519043},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.4284818768501282},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.24207645654678345},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.05191037058830261}],"concepts":[{"id":"https://openalex.org/C140142295","wikidata":"https://www.wikidata.org/wiki/Q617417","display_name":"Isoperimetric inequality","level":2,"score":0.9317944049835205},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7890989780426025},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.7314302921295166},{"id":"https://openalex.org/C164866538","wikidata":"https://www.wikidata.org/wiki/Q367351","display_name":"Cluster (spacecraft)","level":2,"score":0.6335017085075378},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.6145790219306946},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5988266468048096},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.571588397026062},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.502526044845581},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.4541163444519043},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4284818768501282},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.24207645654678345},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.05191037058830261},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3934/nhm.2023053","is_oa":true,"landing_page_url":"https://doi.org/10.3934/nhm.2023053","pdf_url":null,"source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks and Heterogeneous Media","raw_type":"journal-article"},{"id":"pmh:oai:arpi.unipi.it:11568/1213084","is_oa":false,"landing_page_url":"https://hdl.handle.net/11568/1213084","pdf_url":null,"source":{"id":"https://openalex.org/S4377196265","display_name":"CINECA IRIS Institutial research information system (University of Pisa)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I108290504","host_organization_name":"University of Pisa","host_organization_lineage":["https://openalex.org/I108290504"],"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":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:85e5c8797f3c4f7c8af065d8521a0034","is_oa":true,"landing_page_url":"https://doaj.org/article/85e5c8797f3c4f7c8af065d8521a0034","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Networks and Heterogeneous Media, Vol 18, Iss 3, Pp 1226-1235 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3934/nhm.2023053","is_oa":true,"landing_page_url":"https://doi.org/10.3934/nhm.2023053","pdf_url":null,"source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks and Heterogeneous Media","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1606406979","https://openalex.org/W1968647136","https://openalex.org/W1987050938","https://openalex.org/W1996608765","https://openalex.org/W2012137820","https://openalex.org/W2034469935","https://openalex.org/W2049512721","https://openalex.org/W2051150310","https://openalex.org/W2064570349","https://openalex.org/W2067652777","https://openalex.org/W2070632577","https://openalex.org/W2072157555","https://openalex.org/W2101287554","https://openalex.org/W2415568859","https://openalex.org/W2480150893","https://openalex.org/W2564750176","https://openalex.org/W2612390739","https://openalex.org/W2963098465","https://openalex.org/W2964102184","https://openalex.org/W2997045071","https://openalex.org/W3022538125","https://openalex.org/W3087020611","https://openalex.org/W4206388587","https://openalex.org/W4229690478","https://openalex.org/W4280587012","https://openalex.org/W6605955270"],"related_works":["https://openalex.org/W2393012656","https://openalex.org/W4293733602","https://openalex.org/W4387415199","https://openalex.org/W2587882711","https://openalex.org/W2799900090","https://openalex.org/W4297475888","https://openalex.org/W2042006853","https://openalex.org/W4299574858","https://openalex.org/W2047527454","https://openalex.org/W2037470587"],"abstract_inverted_index":{"<abstract><p>In":[0],"this":[1],"paper":[2],"we":[3,154],"study":[4],"infinite":[5,9,159],"isoperimetric":[6,160],"clusters.":[7],"An":[8],"cluster":[10,52,93],"$":[11,13,15,18,26,36,43,53,55,59,62,66,72,99,110,112,114,133,141,144],"{\\bf{E}}":[12,54,143],"in":[14,95],"\\mathbb":[16,28],"R^d":[17,29],"is":[19,47,76],"a":[20,51,92,127],"sequence":[21],"of":[22,33,50,63,107,126,150,158],"disjoint":[23],"measurable":[24],"sets":[25],"E_k\\subset":[27],"$,":[30,68,74,103,139],"called":[31],"regions":[32,65,83],"the":[34,48,64,80,85,96,108,124,131,146,151],"cluster,":[35],"k":[37],"=":[38,101,135],"1,":[39],"2,":[40],"3,":[41],"\\dots":[42],"A":[44],"natural":[45],"question":[46],"existence":[49,125],"with":[56,82,113,130],"given":[57],"volumes":[58],"a_k\\ge":[60],"0":[61],"E_k":[67],"having":[69,84],"finite":[70],"perimeter":[71],"P({\\bf{E}})":[73,134],"which":[75],"minimal":[77],"among":[78],"all":[79],"clusters":[81,161],"same":[86],"volumes.":[87],"We":[88,121],"prove":[89],"that":[90],"such":[91],"exists":[94],"planar":[97],"case":[98],"d":[100],"2":[102],"for":[104,162],"any":[105],"choice":[106],"areas":[109],"a_k":[111,117],"\\sum":[115],"\\sqrt":[116],"<":[118],"\\infty":[119],"$.":[120],"also":[122],"show":[123],"bounded":[128],"minimizer":[129],"property":[132],"\\mathcal":[136],"H^1({\\tilde\\partial}":[137],"{\\bf{E}})":[138],"where":[140],"{\\tilde\\partial}":[142],"denotes":[145],"measure":[147],"theoretic":[148],"boundary":[149],"cluster.":[152],"Finally,":[153],"provide":[155],"several":[156],"examples":[157],"anisotropic":[163],"and":[164],"fractional":[165],"perimeters.</p></abstract>":[166]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3}],"updated_date":"2026-02-26T08:16:20.718346","created_date":"2025-10-10T00:00:00"}