{"id":"https://openalex.org/W2547130799","doi":"https://doi.org/10.1007/s00332-016-9346-1","title":"Sharp $$N^{3/4}$$ N 3 / 4 Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice","display_name":"Sharp $$N^{3/4}$$ N 3 / 4 Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice","publication_year":2016,"publication_date":"2016-11-05","ids":{"openalex":"https://openalex.org/W2547130799","doi":"https://doi.org/10.1007/s00332-016-9346-1","mag":"2547130799","pmid":"https://pubmed.ncbi.nlm.nih.gov/28747811"},"language":"en","primary_location":{"id":"doi:10.1007/s00332-016-9346-1","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-016-9346-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-016-9346-1.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00332-016-9346-1.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006393097","display_name":"Elisa Davoli","orcid":"https://orcid.org/0000-0002-1715-5004"},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Elisa Davoli","raw_affiliation_strings":["Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria","Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria","institution_ids":["https://openalex.org/I129774422"]},{"raw_affiliation_string":"Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria","institution_ids":["https://openalex.org/I129774422"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073587606","display_name":"Paolo Piovano","orcid":"https://orcid.org/0000-0002-4860-7907"},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Paolo Piovano","raw_affiliation_strings":["Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria","Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria"],"raw_orcid":"https://orcid.org/0000-0002-4860-7907","affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria","institution_ids":["https://openalex.org/I129774422"]},{"raw_affiliation_string":"Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria","institution_ids":["https://openalex.org/I129774422"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5087439841","display_name":"Ulisse Stefanelli","orcid":"https://orcid.org/0000-0002-3969-2716"},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]},{"id":"https://openalex.org/I4210137033","display_name":"Istituto di Matematica Applicata e Tecnologie Informatiche","ror":"https://ror.org/03m0n3c07","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210137033","https://openalex.org/I4210155236"]}],"countries":["AT","IT"],"is_corresponding":false,"raw_author_name":"Ulisse Stefanelli","raw_affiliation_strings":["Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria","Istituto di Matematica Applicata e Tecnologie Informatiche \"E. Magenes\" - CNR, v. Ferrata 1, 27100 Pavia, Italy","Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria","institution_ids":["https://openalex.org/I129774422"]},{"raw_affiliation_string":"Istituto di Matematica Applicata e Tecnologie Informatiche \"E. Magenes\" - CNR, v. Ferrata 1, 27100 Pavia, Italy","institution_ids":["https://openalex.org/I4210137033"]},{"raw_affiliation_string":"Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria","institution_ids":["https://openalex.org/I129774422"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5073587606"],"corresponding_institution_ids":["https://openalex.org/I129774422"],"apc_list":{"value":3290,"currency":"USD","value_usd":3290},"apc_paid":{"value":3290,"currency":"USD","value_usd":3290},"fwci":1.4771,"has_fulltext":true,"cited_by_count":23,"citation_normalized_percentile":{"value":0.81338261,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"27","issue":"2","first_page":"627","last_page":"660"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9958999752998352,"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"}},{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9915000200271606,"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":0.9856075048446655},{"id":"https://openalex.org/keywords/perimeter","display_name":"Perimeter","score":0.7737475633621216},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7065528035163879},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.686677098274231},{"id":"https://openalex.org/keywords/hexagonal-lattice","display_name":"Hexagonal lattice","score":0.5973672866821289},{"id":"https://openalex.org/keywords/hexagonal-crystal-system","display_name":"Hexagonal crystal system","score":0.5631140470504761},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.5402683019638062},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.4923769235610962},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.46885451674461365},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.46448999643325806},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.4406660199165344},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.41070687770843506},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3727450966835022},{"id":"https://openalex.org/keywords/condensed-matter-physics","display_name":"Condensed matter physics","score":0.1929459273815155},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.18793156743049622},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.15025809407234192},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.07327467203140259},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06843730807304382}],"concepts":[{"id":"https://openalex.org/C140142295","wikidata":"https://www.wikidata.org/wiki/Q617417","display_name":"Isoperimetric inequality","level":2,"score":0.9856075048446655},{"id":"https://openalex.org/C98503990","wikidata":"https://www.wikidata.org/wiki/Q28474","display_name":"Perimeter","level":2,"score":0.7737475633621216},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7065528035163879},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.686677098274231},{"id":"https://openalex.org/C90626534","wikidata":"https://www.wikidata.org/wiki/Q4523381","display_name":"Hexagonal lattice","level":3,"score":0.5973672866821289},{"id":"https://openalex.org/C128765274","wikidata":"https://www.wikidata.org/wiki/Q663314","display_name":"Hexagonal crystal system","level":2,"score":0.5631140470504761},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.5402683019638062},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.4923769235610962},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.46885451674461365},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.46448999643325806},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4406660199165344},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.41070687770843506},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3727450966835022},{"id":"https://openalex.org/C26873012","wikidata":"https://www.wikidata.org/wiki/Q214781","display_name":"Condensed matter physics","level":1,"score":0.1929459273815155},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.18793156743049622},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.15025809407234192},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.07327467203140259},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06843730807304382},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","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/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0},{"id":"https://openalex.org/C155355069","wikidata":"https://www.wikidata.org/wiki/Q575224","display_name":"Antiferromagnetism","level":2,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1007/s00332-016-9346-1","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-016-9346-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-016-9346-1.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},{"id":"pmid:28747811","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/28747811","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":"Journal of nonlinear science","raw_type":"Journal Article"},{"id":"pmh:oai:re.public.polimi.it:11311/1175260","is_oa":true,"landing_page_url":"http://hdl.handle.net/11311/1175260","pdf_url":"http://hdl.handle.net/11311/1175260","source":{"id":"https://openalex.org/S4306400312","display_name":"Virtual Community of Pathological Anatomy (University of Castilla La Mancha)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79189158","host_organization_name":"University of Castilla-La Mancha","host_organization_lineage":["https://openalex.org/I79189158"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:europepmc.org:4363406","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/5501203","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"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":"pmh:info:fedora/oai:univie.ac.at:o:526565","is_oa":true,"landing_page_url":"https://phaidra.univie.ac.at/o:526565","pdf_url":null,"source":{"id":"https://openalex.org/S4306401993","display_name":"Phaidra (Universit\u00e4t Wien)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I129774422","host_organization_name":"University of Vienna","host_organization_lineage":["https://openalex.org/I129774422"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:0938-8974","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1007/s00332-016-9346-1","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-016-9346-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-016-9346-1.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2610490205","display_name":"Center for Nonlinear Analysis: Research and Training in Applied Mathematics","funder_award_id":"0635983","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2998061923","display_name":"Variational Modeling of Carbon Nanostructures","funder_award_id":"MA14-009","funder_id":"https://openalex.org/F4320321003","funder_display_name":"Vienna Science and Technology Fund"},{"id":"https://openalex.org/G3402382677","display_name":null,"funder_award_id":"0967140","funder_id":"https://openalex.org/F4320337370","funder_display_name":"Office of International Science and Engineering"},{"id":"https://openalex.org/G385473759","display_name":"Variationals Methods in Imaging and in Materials","funder_award_id":"0905778","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5460264237","display_name":null,"funder_award_id":"27052","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G6225531309","display_name":"Variational Structures in Thermomechanics of Solids","funder_award_id":"I2375","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G7041476167","display_name":"PIRE: Science at the Triple Point Between Mathematics, Mechanics and Materials Science","funder_award_id":"0967140","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8799912376","display_name":"Global variational methods for nonlinear evolution","funder_award_id":"P 27052","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G891797142","display_name":null,"funder_award_id":"OISE-0967140","funder_id":"https://openalex.org/F4320337370","funder_display_name":"Office of International Science and Engineering"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320321003","display_name":"Vienna Science and Technology Fund","ror":"https://ror.org/01f9mc681"},{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"},{"id":"https://openalex.org/F4320321691","display_name":"Universit\u00e4t Wien","ror":"https://ror.org/03prydq77"},{"id":"https://openalex.org/F4320337370","display_name":"Office of International Science and Engineering","ror":"https://ror.org/01k638r21"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2547130799.pdf","grobid_xml":"https://content.openalex.org/works/W2547130799.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W1505597232","https://openalex.org/W1522382323","https://openalex.org/W1581091853","https://openalex.org/W1861295442","https://openalex.org/W1978220498","https://openalex.org/W1990829398","https://openalex.org/W1997105089","https://openalex.org/W2023085269","https://openalex.org/W2025633460","https://openalex.org/W2029545793","https://openalex.org/W2062926473","https://openalex.org/W2082852204","https://openalex.org/W2137595244","https://openalex.org/W2139767033","https://openalex.org/W2148905074","https://openalex.org/W2152468302","https://openalex.org/W2514662701","https://openalex.org/W2963127802","https://openalex.org/W3100917591","https://openalex.org/W3104324671","https://openalex.org/W3105309195"],"related_works":["https://openalex.org/W2167212705","https://openalex.org/W2962856485","https://openalex.org/W4386721516","https://openalex.org/W4287597926","https://openalex.org/W2755106943","https://openalex.org/W4299550404","https://openalex.org/W3108476229","https://openalex.org/W3099679575","https://openalex.org/W2756377600","https://openalex.org/W3150576000"],"abstract_inverted_index":{"We":[0],"investigate":[1],"the":[2,12,20,23,27,101],"edge-isoperimetric":[3],"problem":[4],"(EIP)":[5],"for":[6],"sets":[7],"of":[8,22,35,45,70,74,100,103],"n":[9],"points":[10,92],"in":[11,26,68],"triangular":[13],"lattice":[14],"by":[15,86,117],"emphasizing":[16],"its":[17],"relation":[18],"with":[19,89],"emergence":[21],"Wulff":[24],"shape":[25],"crystallization":[28],"problem.":[29],"By":[30,96],"introducing":[31],"a":[32,97],"suitable":[33],"notion":[34],"perimeter":[36,55,64],"and":[37,53,62,129],"area,":[38],"EIP":[39,78],"minimizers":[40,79,107],"are":[41,65,81,109],"characterized":[42],"as":[43],"extremizers":[44],"an":[46],"isoperimetric":[47,76],"inequality:":[48],"they":[49],"attain":[50],"maximal":[51,60],"area":[52,61],"minimal":[54,63],"among":[56],"connected":[57],"configurations.":[58],"The":[59,123],"explicitly":[66,127],"quantified":[67],"terms":[69],"n.":[71],"In":[72],"view":[73],"this":[75],"characterizations,":[77],"$$M_n$$":[80,108],"seen":[82],"to":[83,111,131],"be":[84,132],"given":[85],"hexagonal":[87,115],"configurations":[88,116],"some":[90],"extra":[91,105],"at":[93,118],"their":[94],"boundary.":[95],"careful":[98],"computation":[99],"cardinality":[102],"these":[104],"points,":[106],"estimated":[110],"deviate":[112],"from":[113],"such":[114],"most":[119],"$$K_t\\,":[120],"n^{3/4}+\\mathrm{o}(n^{3/4})$$":[121],"points.":[122],"constant":[124],"$$K_t$$":[125],"is":[126],"determined":[128],"shown":[130],"sharp.":[133]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2}],"updated_date":"2026-08-28T12:50:07.497085","created_date":"2016-11-11T00:00:00"}