{"id":"https://openalex.org/W2978078294","doi":"https://doi.org/10.1007/s00454-020-00259-z","title":"Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes","display_name":"Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes","publication_year":2020,"publication_date":"2020-12-10","ids":{"openalex":"https://openalex.org/W2978078294","doi":"https://doi.org/10.1007/s00454-020-00259-z","mag":"2978078294"},"language":"lv","primary_location":{"id":"doi:10.1007/s00454-020-00259-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00259-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00259-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 &amp; Computational Geometry","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00259-z.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010851411","display_name":"Zakhar Kabluchko","orcid":"https://orcid.org/0000-0001-8483-3373"},"institutions":[{"id":"https://openalex.org/I22465464","display_name":"University of M\u00fcnster","ror":"https://ror.org/00pd74e08","country_code":"DE","type":"education","lineage":["https://openalex.org/I22465464"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Zakhar Kabluchko","raw_affiliation_strings":["Institut f\u00fcr Mathematische Stochastik, Westf\u00e4lische Wilhelms-Universit\u00e4t M\u00fcnster, Orl\u00e9ans-Ring 10, 48149, M\u00fcnster, Germany","Institut f\u00fcr Mathematische Stochastik Westf\u00e4lische Wilhelms-Universit\u00e4t M\u00fcnster, M\u00fcnster, Germany"],"raw_orcid":"https://orcid.org/0000-0001-8483-3373","affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematische Stochastik, Westf\u00e4lische Wilhelms-Universit\u00e4t M\u00fcnster, Orl\u00e9ans-Ring 10, 48149, M\u00fcnster, Germany","institution_ids":["https://openalex.org/I22465464"]},{"raw_affiliation_string":"Institut f\u00fcr Mathematische Stochastik Westf\u00e4lische Wilhelms-Universit\u00e4t M\u00fcnster, M\u00fcnster, Germany","institution_ids":["https://openalex.org/I22465464"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5010851411"],"corresponding_institution_ids":["https://openalex.org/I22465464"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.5847,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.58396412,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"66","issue":"3","first_page":"902","last_page":"937"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9998999834060669,"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":0.9998999834060669,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9764000177383423,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9736999869346619,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.787926435470581},{"id":"https://openalex.org/keywords/simplex","display_name":"Simplex","score":0.7283063530921936},{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.6137968897819519},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.543113648891449},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.48898184299468994},{"id":"https://openalex.org/keywords/convex-hull","display_name":"Convex hull","score":0.44311535358428955},{"id":"https://openalex.org/keywords/beta","display_name":"BETA (programming language)","score":0.4296948313713074},{"id":"https://openalex.org/keywords/convex-body","display_name":"Convex body","score":0.4232999384403229},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.42041856050491333},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.36611998081207275},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3409099876880646}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.787926435470581},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.7283063530921936},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.6137968897819519},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.543113648891449},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.48898184299468994},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.44311535358428955},{"id":"https://openalex.org/C2776174256","wikidata":"https://www.wikidata.org/wiki/Q830842","display_name":"BETA (programming language)","level":2,"score":0.4296948313713074},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.4232999384403229},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.42041856050491333},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.36611998081207275},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3409099876880646},{"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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s00454-020-00259-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00259-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00259-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 &amp; Computational Geometry","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1907.07534","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1907.07534","pdf_url":"https://arxiv.org/pdf/1907.07534","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"mag:2978078294","is_oa":true,"landing_page_url":"http://arxiv.org/pdf/1907.07534.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1907.07534","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1907.07534","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"article"}],"best_oa_location":{"id":"doi:10.1007/s00454-020-00259-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00259-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00259-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 &amp; Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1177974802","display_name":null,"funder_award_id":"EXC 2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G1652504919","display_name":null,"funder_award_id":"EXC 2044","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2110285270","display_name":null,"funder_award_id":"2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2817606592","display_name":null,"funder_award_id":"390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G3237484952","display_name":null,"funder_award_id":"Germany's Excellence Strategy EXC 2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G5074471488","display_name":null,"funder_award_id":"Excellence Strategy EXC 2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8055832297","display_name":null,"funder_award_id":"EXC 2044-390685587, Mathematics M\u00fcnster: Dynamics-Geometry-Structure","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8191055271","display_name":null,"funder_award_id":"EXC 2044 - 390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2978078294.pdf","grobid_xml":"https://content.openalex.org/works/W2978078294.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W48111051","https://openalex.org/W87904750","https://openalex.org/W178783221","https://openalex.org/W565214016","https://openalex.org/W1541118220","https://openalex.org/W1607203414","https://openalex.org/W1975341768","https://openalex.org/W1983121680","https://openalex.org/W1990705237","https://openalex.org/W1993588229","https://openalex.org/W1998315626","https://openalex.org/W2014754855","https://openalex.org/W2015528458","https://openalex.org/W2024312634","https://openalex.org/W2025223969","https://openalex.org/W2029948740","https://openalex.org/W2072369786","https://openalex.org/W2080800043","https://openalex.org/W2087433503","https://openalex.org/W2133006861","https://openalex.org/W2163472056","https://openalex.org/W2319567436","https://openalex.org/W2323065928","https://openalex.org/W2334311386","https://openalex.org/W2605986936","https://openalex.org/W2963457358","https://openalex.org/W2963512376","https://openalex.org/W2963529532","https://openalex.org/W2982807327","https://openalex.org/W2990376029","https://openalex.org/W3040640913","https://openalex.org/W3047298799","https://openalex.org/W3080146149","https://openalex.org/W3097002734","https://openalex.org/W4234349222","https://openalex.org/W4237697075","https://openalex.org/W4239154105","https://openalex.org/W4253948983","https://openalex.org/W6848505933"],"related_works":["https://openalex.org/W3110840479","https://openalex.org/W2599553067","https://openalex.org/W2963558003","https://openalex.org/W2265283073","https://openalex.org/W2888009571","https://openalex.org/W3102288895","https://openalex.org/W2787536425","https://openalex.org/W3026153476","https://openalex.org/W2964068385","https://openalex.org/W2502916942","https://openalex.org/W2167025208","https://openalex.org/W1974343573","https://openalex.org/W2337933709","https://openalex.org/W2964049553","https://openalex.org/W3105206612","https://openalex.org/W2011668878","https://openalex.org/W3098461411","https://openalex.org/W2762868120","https://openalex.org/W3162282390","https://openalex.org/W2600350570"],"abstract_inverted_index":{"Abstract":[0],"Consider":[1],"a":[2],"random":[3,35,178],"simplex":[4,106],"$$[X_1,\\ldots":[5,107,130],",X_n]$$":[6,108],"<mml:math":[7,39,58,81,109,132,154,186,208,286,307],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[8,40,59,82,110,133,155,187,209,287,308],"<mml:mrow>":[9,41,61,64,83,86,92,111,134,156,162,168,210,216,224,231,242,245,254,257,266,268,288,291,297,309],"<mml:mo>[</mml:mo>":[10,112,135],"<mml:msub>":[11,18,42,49,84,113,120,136,143,157,211,289,310],"<mml:mi>X</mml:mi>":[12,19,43,50,114,121,137,144],"<mml:mn>1</mml:mn>":[13,44,67,115,138,219,233,260,271],"</mml:msub>":[14,21,45,52,91,116,123,139,146,167,223,296],"<mml:mo>,</mml:mo>":[15,17,46,48,88,117,119,140,142,164,220,250,263,293],"<mml:mo>\u2026</mml:mo>":[16,47,118,141],"<mml:mi>n</mml:mi>":[20,51,65,87,122,163,217,258,269,292],"<mml:mo>]</mml:mo>":[22,124,147],"</mml:mrow>":[23,53,63,68,90,96,97,125,148,166,172,173,222,228,238,244,248,256,261,272,274,278,295,301,302],"</mml:math>":[24,54,70,98,126,149,174,192,279,303],"defined":[25],"as":[26],"the":[27,72,100,105,183],"convex":[28],"hull":[29],"of":[30,104],"independent":[31],"identically":[32],"distributed":[33,180],"(i.i.d.)":[34],"points":[36,179],"$$X_1,\\ldots":[37],",X_n$$":[38],"in":[55],"$$\\mathbb":[56],"{R}^{n-1}$$":[57],"<mml:msup>":[60,188,230,241,253],"<mml:mi>R</mml:mi>":[62,255],"<mml:mo>-</mml:mo>":[66,218,246,259,270],"</mml:msup>":[69,191,240,249,262],"with":[71],"following":[73],"beta":[74,184],"density:":[75],"\"Equation":[76],"missing\"":[77],"Let":[78],"$$J_{n,k}(\\beta":[79,284],")$$":[80,153,285,306],"<mml:mi>J</mml:mi>":[85,159,290,312],"<mml:mi>k</mml:mi>":[89,145,165,294],"<mml:mo>(</mml:mo>":[93,169,225,232,267,298],"<mml:mi>\u03b2</mml:mi>":[94,170,221,247,264,299],"<mml:mo>)</mml:mo>":[95,171,227,243,273,300],"be":[99],"expected":[101],"internal":[102],"angle":[103],"at":[127],"its":[128],"face":[129],",X_k]$$":[131],".":[150],"Define":[151],"$${\\tilde{J}}_{n,k}(\\beta":[152,305],"<mml:mover>":[158,212,311],"<mml:mo>~</mml:mo>":[160,214,313],"</mml:mover>":[161,215,314],"analogously":[175],"for":[176,283],"i.i.d.":[177],"according":[181],"to":[182],"$$'$$":[185],"<mml:mrow/>":[189],"<mml:mo>\u2032</mml:mo>":[190],"density":[193],"$${\\tilde{f}}_{n-1,\\beta":[194],"}":[195],"(x)":[196],"\\propto":[197],"(1+\\Vert":[198],"x\\Vert":[199],"^2)^{-\\beta":[200],"},":[201],"x\\in":[202],"\\mathbb":[203],"{R}^{n-1},":[204],"\\beta":[205],"&gt;":[206],"({n-1})/{2}.$$":[207],"<mml:mi>f</mml:mi>":[213],"<mml:mi>x</mml:mi>":[226,236,251],"<mml:mo>\u221d</mml:mo>":[229],"<mml:mo>+</mml:mo>":[234],"<mml:mo>\u2016</mml:mo>":[235,237],"<mml:mn>2</mml:mn>":[239,276],"<mml:mo>\u2208</mml:mo>":[252],"<mml:mo>&gt;</mml:mo>":[265],"<mml:mo>/</mml:mo>":[275],"<mml:mo>.</mml:mo>":[277],"We":[280],"derive":[281],"formulae":[282],"and":[304],"<mml:m":[315]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}