{"id":"https://openalex.org/W4327989522","doi":"https://doi.org/10.1007/s00454-025-00780-z","title":"Optimal Volume-Sensitive Bounds for Polytope Approximation","display_name":"Optimal Volume-Sensitive Bounds for Polytope Approximation","publication_year":2025,"publication_date":"2025-09-20","ids":{"openalex":"https://openalex.org/W4327989522","doi":"https://doi.org/10.1007/s00454-025-00780-z"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-025-00780-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-025-00780-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00780-z.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete & Computational Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00780-z.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027296680","display_name":"Sunil K. Arya","orcid":"https://orcid.org/0000-0002-3425-4231"},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Sunil Arya","raw_affiliation_strings":["Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Sai Kung, Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Sai Kung, Hong Kong","institution_ids":["https://openalex.org/I200769079"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016442699","display_name":"David M. Mount","orcid":"https://orcid.org/0000-0002-3290-8932"},"institutions":[{"id":"https://openalex.org/I66946132","display_name":"University of Maryland, College Park","ror":"https://ror.org/047s2c258","country_code":"US","type":"education","lineage":["https://openalex.org/I66946132"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"David M. Mount","raw_affiliation_strings":["Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland, USA"],"raw_orcid":"https://orcid.org/0000-0002-3290-8932","affiliations":[{"raw_affiliation_string":"Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland, USA","institution_ids":["https://openalex.org/I66946132"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5016442699"],"corresponding_institution_ids":["https://openalex.org/I66946132"],"apc_list":{"value":3090,"currency":"USD","value_usd":3090},"apc_paid":{"value":3090,"currency":"USD","value_usd":3090},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00180562,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"74","issue":"4","first_page":"839","last_page":"871"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9988999962806702,"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":0.9988999962806702,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9261999726295471,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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":0.8067269921302795},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7384197115898132},{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.7000443935394287},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.6707540154457092},{"id":"https://openalex.org/keywords/convex-body","display_name":"Convex body","score":0.6120549440383911},{"id":"https://openalex.org/keywords/isoperimetric-inequality","display_name":"Isoperimetric inequality","score":0.6080371141433716},{"id":"https://openalex.org/keywords/ball","display_name":"Ball (mathematics)","score":0.6030482053756714},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5558425784111023},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.5556151270866394},{"id":"https://openalex.org/keywords/surface","display_name":"Surface (topology)","score":0.4205881953239441},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.4173823297023773},{"id":"https://openalex.org/keywords/convex-hull","display_name":"Convex hull","score":0.3254055976867676},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.207520991563797}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8067269921302795},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7384197115898132},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.7000443935394287},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.6707540154457092},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.6120549440383911},{"id":"https://openalex.org/C140142295","wikidata":"https://www.wikidata.org/wiki/Q617417","display_name":"Isoperimetric inequality","level":2,"score":0.6080371141433716},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.6030482053756714},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5558425784111023},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5556151270866394},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.4205881953239441},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.4173823297023773},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.3254055976867676},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.207520991563797}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1007/s00454-025-00780-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-025-00780-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00780-z.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete & Computational Geometry","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2303.09586","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2303.09586","pdf_url":"https://arxiv.org/pdf/2303.09586","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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:drops-oai.dagstuhl.de:17859","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.9","pdf_url":"https://drops.dagstuhl.de/storage/00lipics/lipics-vol258-socg2023/LIPIcs.SoCG.2023.9/LIPIcs.SoCG.2023.9.pdf","source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"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":"InProceedings"},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-127306","is_oa":false,"landing_page_url":"http://repository.hkust.edu.hk/ir/Record/1783.1-127306","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"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":"Conference paper"},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-138991","is_oa":false,"landing_page_url":"http://repository.hkust.edu.hk/ir/Record/1783.1-138991","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Conference paper"},{"id":"doi:10.48550/arxiv.2303.09586","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2303.09586","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.4230/lipics.socg.2023.9","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.socg.2023.9","pdf_url":null,"source":{"id":"https://openalex.org/S4393917817","display_name":"Leibniz international proceedings in informatics","issn_l":"1868-8969","issn":["1868-8969"],"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"ConferencePaper"}],"best_oa_location":{"id":"doi:10.1007/s00454-025-00780-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-025-00780-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00780-z.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete & Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2348294594","display_name":null,"funder_award_id":"16214721","funder_id":"https://openalex.org/F4320329815","funder_display_name":"Hong Kong Government"},{"id":"https://openalex.org/G7169651934","display_name":null,"funder_award_id":"16213219","funder_id":"https://openalex.org/F4320329815","funder_display_name":"Hong Kong Government"}],"funders":[{"id":"https://openalex.org/F4320329815","display_name":"Hong Kong Government","ror":"https://ror.org/034179816"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4327989522.pdf","grobid_xml":"https://content.openalex.org/works/W4327989522.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W2806113084","https://openalex.org/W2247953682","https://openalex.org/W4295128493","https://openalex.org/W2063567013","https://openalex.org/W2137746155","https://openalex.org/W7111320267","https://openalex.org/W3098818037","https://openalex.org/W4401189978","https://openalex.org/W2037318282","https://openalex.org/W2496539891","https://openalex.org/W2154952480","https://openalex.org/W2952392285","https://openalex.org/W2075603209","https://openalex.org/W1995232397","https://openalex.org/W2086591901","https://openalex.org/W2006109604","https://openalex.org/W2052967435","https://openalex.org/W4248484045","https://openalex.org/W4210733153","https://openalex.org/W2738786117","https://openalex.org/W4399276472","https://openalex.org/W2316432841","https://openalex.org/W119189535","https://openalex.org/W4241014687","https://openalex.org/W4412215459","https://openalex.org/W596522316","https://openalex.org/W2945305319","https://openalex.org/W2047308801","https://openalex.org/W2070615572","https://openalex.org/W1993588229","https://openalex.org/W1971662030","https://openalex.org/W2021445627","https://openalex.org/W4205296686","https://openalex.org/W3024390163","https://openalex.org/W117977885","https://openalex.org/W2502358513","https://openalex.org/W1979306594","https://openalex.org/W4290464992","https://openalex.org/W2083027876"],"related_works":["https://openalex.org/W1970960579","https://openalex.org/W2402215390","https://openalex.org/W2969915209","https://openalex.org/W4297581286","https://openalex.org/W2952368903","https://openalex.org/W4296580083","https://openalex.org/W2949500586","https://openalex.org/W2371177213","https://openalex.org/W1867657303","https://openalex.org/W1975735936"],"abstract_inverted_index":{"Abstract":[0],"Approximating":[1],"convex":[2,19,99,110,124],"bodies":[3],"is":[4,33,54,93,112,138,167],"a":[5,12,18,45,109,123,145],"fundamental":[6],"question":[7],"in":[8,22,113],"geometry,":[9],"which":[10],"has":[11],"wide":[13],"variety":[14],"of":[15,38,40,108,115,144,148,177],"applications.":[16],"Given":[17,122],"body":[20,125],"K":[21,64,126,136,153,165,191,199,205,242],"$$\\mathbb":[23],"{R}^d$$":[24],"R":[25],"d":[26,29,70,134,161,195,203,238,248],"for":[27,44,82,97,170],"fixed":[28],",":[30,127],"the":[31,36,87,106,119,142,149,178,215,223,227],"objective":[32],"to":[34,104,118,140,222,230],"minimize":[35],"number":[37],"facets":[39,77],"an":[41],"approximating":[42],"polytope":[43],"given":[46],"Hausdorff":[47],"error":[48],"$$\\varepsilon":[49],"$$":[50],"\u03b5":[51,67,245],".":[52,154,207],"It":[53,173],"known":[55],"that":[56,181,214],"$$O(({{\\,\\textrm{diam}\\,}}(K)/\\varepsilon":[57],")^{(d-1)/2})$$":[58,233],"O":[59,234],"(":[60,61,63,69,135,164,190,198,204,235,236,241,247],"diam":[62,189],")":[65,68,73,76,137,166,192,200,206,243,246,251],"/":[66,74,244],"-":[71,162,196,239,249],"1":[72,163,197,240,250],"2":[75],"suffice":[78],"and":[79,211],"are":[80],"necessary":[81],"many":[83],"instances,":[84],"such":[85],"as":[86,152],"Euclidean":[88,120,146],"ball.":[89,121],"However,":[90],"this":[91],"bound":[92,217,229],"far":[94],"from":[95,175],"optimal":[96],"\u201cskinny\u201d":[98],"bodies.":[100],"A":[101],"natural":[102],"way":[103],"characterize":[105],"skinniness":[107],"object":[111],"terms":[114],"its":[116,128],"relationship":[117],"volume":[129,151],"diameter":[130,143,157],"$$\\Delta":[131,158],"_d(K)$$":[132,188],"\u0394":[133,160,194,202,237],"defined":[139,168],"be":[141,219],"ball":[147],"same":[150],"The":[155],"surface":[156,171,224],"_{d-1}(K)$$":[159],"analogously":[169],"area.":[172],"follows":[174],"generalizations":[176],"isoperimetric":[179],"inequality":[180],"$${{\\,\\textrm{diam}\\,}}(K)":[182],"\\ge":[183,186],"\\Delta":[184,187],"_{d-1}(K)":[185],"\u2265":[193,201],"Arya,":[208],"da":[209],"Fonseca,":[210],"Mount":[212],"proved":[213],"diameter-based":[216],"could":[218],"made":[220],"sensitive":[221],"diameter,":[225],"improving":[226],"above":[228],"$$O((\\Delta":[231],"_{d-1}(K)/\\varepsilon":[232]},"counts_by_year":[],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2023-03-21T00:00:00"}