{"id":"https://openalex.org/W2025239200","doi":"https://doi.org/10.3390/sym5010047","title":"A Note on Lower Bounds for Colourful Simplicial Depth","display_name":"A Note on Lower Bounds for Colourful Simplicial Depth","publication_year":2013,"publication_date":"2013-01-07","ids":{"openalex":"https://openalex.org/W2025239200","doi":"https://doi.org/10.3390/sym5010047","mag":"2025239200"},"language":"en","primary_location":{"id":"doi:10.3390/sym5010047","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym5010047","pdf_url":"https://www.mdpi.com/2073-8994/5/1/47/pdf?version=1357562179","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/5/1/47/pdf?version=1357562179","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004734694","display_name":"Antoine Deza","orcid":"https://orcid.org/0000-0002-2392-4607"},"institutions":[{"id":"https://openalex.org/I98251732","display_name":"McMaster University","ror":"https://ror.org/02fa3aq29","country_code":"CA","type":"education","lineage":["https://openalex.org/I98251732"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Antoine Deza","raw_affiliation_strings":["Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada"],"affiliations":[{"raw_affiliation_string":"Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada","institution_ids":["https://openalex.org/I98251732"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043510929","display_name":"Tamon Stephen","orcid":"https://orcid.org/0000-0003-0156-7415"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Tamon Stephen","raw_affiliation_strings":["Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada","institution_ids":["https://openalex.org/I18014758"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073850100","display_name":"Feng Xie","orcid":"https://orcid.org/0000-0002-8486-0117"},"institutions":[{"id":"https://openalex.org/I98251732","display_name":"McMaster University","ror":"https://ror.org/02fa3aq29","country_code":"CA","type":"education","lineage":["https://openalex.org/I98251732"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Feng Xie","raw_affiliation_strings":["Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada"],"affiliations":[{"raw_affiliation_string":"Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada","institution_ids":["https://openalex.org/I98251732"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5004734694"],"corresponding_institution_ids":["https://openalex.org/I98251732"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.2402,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.60394994,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"5","issue":"1","first_page":"47","last_page":"53"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9742000102996826,"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"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9742000102996826,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9710000157356262,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9383000135421753,"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/combinatorics","display_name":"Combinatorics","score":0.8162548542022705},{"id":"https://openalex.org/keywords/convex-hull","display_name":"Convex hull","score":0.6462925672531128},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6258389949798584},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.6166017651557922},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5780928134918213},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5510568618774414},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"Simplicial complex","score":0.5086320638656616},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.46322229504585266},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.45288845896720886},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2154843807220459},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.14217650890350342},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.06128469109535217}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8162548542022705},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.6462925672531128},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6258389949798584},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6166017651557922},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5780928134918213},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5510568618774414},{"id":"https://openalex.org/C187929450","wikidata":"https://www.wikidata.org/wiki/Q994399","display_name":"Simplicial complex","level":2,"score":0.5086320638656616},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.46322229504585266},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.45288845896720886},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2154843807220459},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.14217650890350342},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.06128469109535217},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym5010047","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym5010047","pdf_url":"https://www.mdpi.com/2073-8994/5/1/47/pdf?version=1357562179","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.361.4289","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.361.4289","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.mdpi.com/2073-8994/5/1/47/pdf?y=1","raw_type":"text"},{"id":"pmh:oai:doaj.org/article:3d47f16a1dc841acb829df847b5cb7fa","is_oa":true,"landing_page_url":"https://doaj.org/article/3d47f16a1dc841acb829df847b5cb7fa","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 5, Iss 1, Pp 47-53 (2013)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym5010047","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym5010047","pdf_url":"https://www.mdpi.com/2073-8994/5/1/47/pdf?version=1357562179","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","score":0.6700000166893005,"id":"https://metadata.un.org/sdg/11"}],"awards":[{"id":"https://openalex.org/G2165548363","display_name":null,"funder_award_id":"Canada","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G3969759488","display_name":null,"funder_award_id":"program","funder_id":"https://openalex.org/F4320322675","funder_display_name":"Mitacs"},{"id":"https://openalex.org/G5784215521","display_name":null,"funder_award_id":"Chair","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G8284766523","display_name":null,"funder_award_id":"(NSERC)","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G8595721295","display_name":null,"funder_award_id":"NSERC","funder_id":"https://openalex.org/F4320320994","funder_display_name":"Canada Research Chairs"},{"id":"https://openalex.org/G919846122","display_name":null,"funder_award_id":"Chair","funder_id":"https://openalex.org/F4320320994","funder_display_name":"Canada Research Chairs"}],"funders":[{"id":"https://openalex.org/F4320320994","display_name":"Canada Research Chairs","ror":"https://ror.org/0517h6h17"},{"id":"https://openalex.org/F4320322675","display_name":"Mitacs","ror":"https://ror.org/00cjrc276"},{"id":"https://openalex.org/F4320323808","display_name":"Universidad de Cantabria","ror":"https://ror.org/046ffzj20"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2025239200.pdf","grobid_xml":"https://content.openalex.org/works/W2025239200.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W1863203090","https://openalex.org/W1973744648","https://openalex.org/W1984879308","https://openalex.org/W1994709142","https://openalex.org/W2001223605","https://openalex.org/W2015695027","https://openalex.org/W2016542791","https://openalex.org/W2022056727","https://openalex.org/W2049678150","https://openalex.org/W2067339112","https://openalex.org/W2089096000","https://openalex.org/W2103307323","https://openalex.org/W2171698682","https://openalex.org/W2293156574","https://openalex.org/W2579687430","https://openalex.org/W3098813812","https://openalex.org/W6697377011","https://openalex.org/W6732865510"],"related_works":["https://openalex.org/W2363904277","https://openalex.org/W1982962927","https://openalex.org/W3157463780","https://openalex.org/W2128601753","https://openalex.org/W2048874717","https://openalex.org/W2372035979","https://openalex.org/W4206371202","https://openalex.org/W2393489669","https://openalex.org/W2375652826","https://openalex.org/W4303635220"],"abstract_inverted_index":{"The":[0],"colourful":[1,41],"simplicial":[2],"depth":[3],"problem":[4],"in":[5,25,36,102],"dimension":[6,103],"d":[7,72,82],"is":[8,23,58,61,85],"to":[9,63,71,97,107],"find":[10],"a":[11,37,94],"configuration":[12],"of":[13,16,29,40],"(d+1)":[14,17],"sets":[15],"points":[18],"such":[19],"that":[20],"the":[21,26,76,99],"origin":[22],"contained":[24,35],"convex":[27],"hull":[28],"each":[30,49],"set,":[31],"or":[32],"colour,":[33],"but":[34],"minimal":[38],"number":[39],"simplices":[42,57],"generated":[43],"by":[44],"taking":[45],"one":[46],"point":[47],"from":[48,105],"set.":[50],"A":[51],"construction":[52],"attaining":[53],"d2":[54],"+":[55],"1":[56],"known,":[59],"and":[60],"conjectured":[62],"be":[64],"minimal.":[65],"This":[66],"has":[67],"been":[68],"confirmed":[69],"up":[70],"=":[73],"3,":[74],"however":[75],"best":[77],"known":[78],"lower":[79,100],"bound":[80,101],"for":[81],"\u2265":[83],"4":[84,104],"\u2308(d+1)2":[86],"/2":[87],"\u2309.":[88],"In":[89],"this":[90],"note,":[91],"we":[92],"use":[93],"branching":[95],"strategy":[96],"improve":[98],"13":[106],"14.":[108]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}