{"id":"https://openalex.org/W2137718369","doi":"https://doi.org/10.1145/1810959.1810996","title":"Distance <i>k</i> -sectors exist","display_name":"Distance <i>k</i> -sectors exist","publication_year":2010,"publication_date":"2010-06-13","ids":{"openalex":"https://openalex.org/W2137718369","doi":"https://doi.org/10.1145/1810959.1810996","mag":"2137718369"},"language":"en","primary_location":{"id":"doi:10.1145/1810959.1810996","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1810959.1810996","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-sixth annual symposium on Computational geometry","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100809646","display_name":"Keiko Imai","orcid":null},"institutions":[{"id":"https://openalex.org/I96679780","display_name":"Chuo University","ror":"https://ror.org/03qvqb743","country_code":"JP","type":"education","lineage":["https://openalex.org/I96679780"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Keiko Imai","raw_affiliation_strings":["Chuo University, Tokyo, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Chuo University, Tokyo, Japan","institution_ids":["https://openalex.org/I96679780"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074573236","display_name":"Akitoshi Kawamura","orcid":"https://orcid.org/0009-0006-3706-7470"},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Akitoshi Kawamura","raw_affiliation_strings":["University of Toronto, Toronto, ON, Canada","University of Toronto, Toronto, On, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Toronto, Toronto, ON, Canada","institution_ids":["https://openalex.org/I185261750"]},{"raw_affiliation_string":"University of Toronto, Toronto, On, Canada","institution_ids":["https://openalex.org/I185261750"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070164050","display_name":"Ji\u0159\u0131\u0301 Matou\u0161ek","orcid":"https://orcid.org/0000-0002-9447-0366"},"institutions":[{"id":"https://openalex.org/I21250087","display_name":"Charles University","ror":"https://ror.org/024d6js02","country_code":"CZ","type":"education","lineage":["https://openalex.org/I21250087"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Ji\u0159i Matou\u0161ek","raw_affiliation_strings":["Charles University, Prague, Czech Rep"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Charles University, Prague, Czech Rep","institution_ids":["https://openalex.org/I21250087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017467848","display_name":"Daniel Reem","orcid":"https://orcid.org/0000-0003-3190-2720"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Daniel Reem","raw_affiliation_strings":["The Technion, Haifa, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"The Technion, Haifa, Israel","institution_ids":["https://openalex.org/I174306211"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031520565","display_name":"Takeshi Tokuyama","orcid":"https://orcid.org/0000-0002-9400-8729"},"institutions":[{"id":"https://openalex.org/I201537933","display_name":"Tohoku University","ror":"https://ror.org/01dq60k83","country_code":"JP","type":"education","lineage":["https://openalex.org/I201537933"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Takeshi Tokuyama","raw_affiliation_strings":["Tohoku University, Sendai, Japan","Tohoku University Sendai, Japan#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tohoku University, Sendai, Japan","institution_ids":["https://openalex.org/I201537933"]},{"raw_affiliation_string":"Tohoku University Sendai, Japan#TAB#","institution_ids":["https://openalex.org/I201537933"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.7211,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.81115353,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"210","last_page":"215"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9997000098228455,"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/T12923","display_name":"Digital Image Processing Techniques","score":0.9950000047683716,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9921000003814697,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.7490626573562622},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7390313744544983},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.645239531993866},{"id":"https://openalex.org/keywords/euclidean-distance","display_name":"Euclidean distance","score":0.6388399004936218},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.6131924390792847},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.5940713882446289},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5569615960121155},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5215911269187927},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.49163228273391724},{"id":"https://openalex.org/keywords/geodesic","display_name":"Geodesic","score":0.4365423023700714},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.43543189764022827},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.4219825565814972},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.4147627353668213},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12241590023040771},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10099667310714722},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09341982007026672}],"concepts":[{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.7490626573562622},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7390313744544983},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.645239531993866},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.6388399004936218},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.6131924390792847},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.5940713882446289},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5569615960121155},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5215911269187927},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.49163228273391724},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.4365423023700714},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.43543189764022827},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.4219825565814972},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4147627353668213},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12241590023040771},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10099667310714722},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09341982007026672},{"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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1810959.1810996","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1810959.1810996","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-sixth annual symposium on Computational geometry","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"},{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W1969554889","https://openalex.org/W1989486331","https://openalex.org/W2041114139","https://openalex.org/W2046213250","https://openalex.org/W2049676311","https://openalex.org/W2101709452","https://openalex.org/W2108043521","https://openalex.org/W2161496261","https://openalex.org/W2570492632","https://openalex.org/W3105764362"],"related_works":["https://openalex.org/W4256429076","https://openalex.org/W1971174658","https://openalex.org/W2099195351","https://openalex.org/W1978591761","https://openalex.org/W4287661821","https://openalex.org/W3133864228","https://openalex.org/W2941285539","https://openalex.org/W2169691698","https://openalex.org/W2591753903","https://openalex.org/W2105228010"],"abstract_inverted_index":{"The":[0,156],"bisector":[1,53],"of":[2,16,30,54,98,109,121,142,158,165],"two":[3,131],"nonempty":[4],"sets":[5,135],"P":[6,23,31,68,79,136,168],"and":[7,24,32,56,69,80,92,107,128,137,169,193],"Q":[8,81,138],"in":[9,84,100,113,139,149,174],"a":[10,42,96,122,162,188,196],"metric":[11,177],"space":[12],"is":[13,38,41,51,161,179],"the":[14,52,76,105,110,119,159,182],"set":[15],"all":[17,126],"points":[18,83],"with":[19],"equal":[20],"distance":[21,28,111,123],"to":[22,25],"Q.":[26,72],"A":[27],"k-sector":[29,124],"Q,":[33,170],"where":[34,65,78],"k":[35,127],"\u2265":[36],"2":[37],"an":[39,175],"integer,":[40],"(k-1)-tuple":[43],"(C1,":[44],"C2,":[45],"...,":[46,63],"Ck-1)":[47],"such":[48],"that":[49],"Ci":[50],"Ci-1":[55],"Ci+1":[57],"for":[58,75,125,129,167,195],"every":[59,130],"i=":[60],"1,":[61],"2,":[62],"k-1,":[64],"C0":[66],"=":[67,71],"Ck":[70],"This":[73],"notion,":[74],"case":[77],"are":[82],"Euclidean":[85,140],"plane,":[86],"was":[87],"introduced":[88,190],"by":[89,95,187,191],"Asano,":[90],"Matousek,":[91],"Tokuyama,":[93],"motivated":[94],"question":[97],"Murata":[99],"VLSI":[101],"design.":[102],"They":[103],"established":[104],"existence":[106,120,172],"uniqueness":[108],"trisector":[112],"this":[114],"special":[115],"case.":[116],"We":[117],"prove":[118],"disjoint,":[132],"nonempty,":[133],"closed":[134],"spaces":[141,152],"any":[143],"(finite)":[144],"dimension,":[145],"or":[146],"more":[147],"generally,":[148],"proper":[150],"geodesic":[151],"(uniqueness":[153],"remains":[154],"open).":[155],"core":[157],"proof":[160],"new":[163],"notion":[164],"k-gradation":[166],"whose":[171],"(even":[173],"arbitrary":[176],"space)":[178],"proved":[180],"using":[181],"Knaster-Tarski":[183],"fixed":[184],"point":[185],"theorem,":[186],"method":[189],"Reem":[192],"Reich":[194],"slightly":[197],"different":[198],"purpose.":[199]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}