{"id":"https://openalex.org/W7213896485","doi":"https://doi.org/10.48550/arxiv.2609.21571","title":"Angles and volumes of simplices, boxes, and crosspolytopes in constant curvature","display_name":"Angles and volumes of simplices, boxes, and crosspolytopes in constant curvature","publication_year":2026,"publication_date":"2026-09-18","ids":{"openalex":"https://openalex.org/W7213896485","doi":"https://doi.org/10.48550/arxiv.2609.21571"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.21571","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21571","pdf_url":"https://arxiv.org/pdf/2609.21571","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,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.21571","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","observed_orcids":["https://orcid.org/0000-0001-8483-3373"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kabluchko, Zakhar","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5119710041","display_name":"Philipp Schange","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Schange, Philipp","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.7089999914169312,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12504","display_name":"Mathematics and Applications","score":0.7089999914169312,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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":0.10419999808073044,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.10279999673366547,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/simplices","display_name":"simplices","score":0.9010000228881836},{"id":"https://openalex.org/keywords/boxes","display_name":"boxes","score":0.9549999833106995},{"id":"https://openalex.org/keywords/spaces-of-constant-curvature","display_name":"spaces of constant curvature","score":0.781000018119812},{"id":"https://openalex.org/keywords/spherical-geometry","display_name":"spherical geometry","score":0.531000018119812},{"id":"https://openalex.org/keywords/hyperbolic-geometry","display_name":"hyperbolic geometry","score":0.9929999709129333},{"id":"https://openalex.org/keywords/internal-angles","display_name":"internal angles","score":0.5799999833106995},{"id":"https://openalex.org/keywords/regular-polytopes","display_name":"regular polytopes","score":0.6660000085830688},{"id":"https://openalex.org/keywords/volume-equations","display_name":"volume equations","score":0.718999981880188},{"id":"https://openalex.org/keywords/integral-representation","display_name":"integral representation","score":0.375},{"id":"https://openalex.org/keywords/normal-distribution-function","display_name":"normal distribution function","score":0.7820000052452087}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7617999911308289},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.6319000124931335},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6183000206947327},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.5605999827384949},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.5343999862670898},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5246999859809875},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5131999850273132},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.4896000027656555},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.48579999804496765},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.4307999908924103},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.42879998683929443},{"id":"https://openalex.org/C135981907","wikidata":"https://www.wikidata.org/wiki/Q188056","display_name":"Triangulation","level":2,"score":0.37599998712539673},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.36970001459121704},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.36320000886917114},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3495999872684479},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.3424000144004822},{"id":"https://openalex.org/C20556612","wikidata":"https://www.wikidata.org/wiki/Q4469374","display_name":"Volume (thermodynamics)","level":2,"score":0.3395000100135803},{"id":"https://openalex.org/C206352148","wikidata":"https://www.wikidata.org/wiki/Q209306","display_name":"Hyperbolic geometry","level":3,"score":0.3395000100135803},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.33160001039505005},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.3158999979496002},{"id":"https://openalex.org/C83677898","wikidata":"https://www.wikidata.org/wiki/Q1878538","display_name":"Hyperbolic space","level":2,"score":0.29989999532699585},{"id":"https://openalex.org/C181104567","wikidata":"https://www.wikidata.org/wiki/Q761383","display_name":"Riemannian geometry","level":2,"score":0.29670000076293945},{"id":"https://openalex.org/C157498722","wikidata":"https://www.wikidata.org/wiki/Q34929","display_name":"Three-dimensional space","level":2,"score":0.2849000096321106},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.27559998631477356},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.27379998564720154},{"id":"https://openalex.org/C153658351","wikidata":"https://www.wikidata.org/wiki/Q746264","display_name":"Constant function","level":3,"score":0.2563000023365021}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.21571","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21571","pdf_url":"https://arxiv.org/pdf/2609.21571","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,"listed_in":[],"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":"doi:10.48550/arxiv.2609.21571","doi":"https://doi.org/10.48550/arxiv.2609.21571","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.21571","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,"listed_in":[],"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":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.21571","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21571","pdf_url":"https://arxiv.org/pdf/2609.21571","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,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"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/G2817606592","display_name":"EXC 2044: Mathematik M\u00fcnster: Dynamik \u2013 Geometrie \u2013 Struktur","funder_award_id":"390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G3617462173","display_name":null,"funder_award_id":"SPP 2265","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/W7213896485.pdf","grobid_xml":"https://content.openalex.org/works/W7213896485.grobid-xml"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W120618969","https://openalex.org/W140961771","https://openalex.org/W1510365205","https://openalex.org/W1546646004","https://openalex.org/W1970082954","https://openalex.org/W1970966506","https://openalex.org/W1985426635","https://openalex.org/W2010669260","https://openalex.org/W2016166616","https://openalex.org/W2016299362","https://openalex.org/W2036203998","https://openalex.org/W2048220644","https://openalex.org/W2072687048","https://openalex.org/W2078808043","https://openalex.org/W2084063180","https://openalex.org/W2095041160","https://openalex.org/W2144006746","https://openalex.org/W2163916252","https://openalex.org/W2605986936","https://openalex.org/W2729236849","https://openalex.org/W3036327512","https://openalex.org/W4210631977","https://openalex.org/W4240386814","https://openalex.org/W4288935732","https://openalex.org/W4405022869","https://openalex.org/W4414448316","https://openalex.org/W7116802689"],"related_works":["https://openalex.org/W4416098267","https://openalex.org/W4414448316","https://openalex.org/W7223091338","https://openalex.org/W2394061741","https://openalex.org/W1977997247","https://openalex.org/W2329271446","https://openalex.org/W7099017656","https://openalex.org/W3036327512","https://openalex.org/W2350187249","https://openalex.org/W762422858"],"abstract_inverted_index":{"We":[0,70],"determine,":[1],"up":[2],"to":[3,156],"isometry,":[4],"the":[5,60,99,124,148,161,167,190,193,198,208],"Riemannian":[6],"tangent":[7],"and":[8,21,44,58,78,108,132,135,142,173],"normal":[9,169],"cones":[10,83],"at":[11,84,176],"arbitrary":[12,48],"faces":[13],"of":[14,26,63,151,192,202],"boxes,":[15,114],"acute":[16],"orthocentric":[17],"simplices,":[18,20,56],"rectangular":[19],"asymmetric":[22,140],"crosspolytopes":[23,109,153],"in":[24,47,67,110,120,160,197],"spaces":[25],"constant":[27],"sectional":[28],"curvature":[29,203],"$\u03ba\\in\\mathbb":[30],"R$.":[31],"Using":[32],"these":[33],"descriptions,":[34],"we":[35,102,115],"derive":[36,116],"explicit":[37],"one-dimensional":[38,144],"integral":[39,145,211],"formulas":[40,97,105,118,129,138,165],"for":[41,106,130,139,147],"their":[42,88],"internal":[43,149],"external":[45],"angles":[46,150],"dimension.":[49,69,122],"In":[50],"particular,":[51],"our":[52,95],"results":[53,126],"cover":[54],"regular":[55,64,76,152,195],"cubes,":[57],"crosspolytopes,":[59,136,141],"three":[61],"families":[62],"polytopes":[65],"existing":[66],"every":[68,121],"also":[71],"establish":[72],"a":[73,143],"correspondence":[74],"between":[75],"simplices":[77,107],"cubes":[79,134],"with":[80,98],"isometric":[81],"local":[82],"corresponding":[85],"faces,":[86],"provided":[87],"edge-length":[89],"parameters":[90],"are":[91,127],"suitably":[92],"matched.":[93],"Combining":[94],"angle":[96,128],"Poincar\u00e9":[100],"relation,":[101],"obtain":[103],"volume":[104,117,137,191],"even":[111,159],"dimensions.":[112],"For":[113,182],"valid":[119],"Among":[123],"new":[125,158],"spherical":[131],"hyperbolic":[133,200],"representation":[146],"that":[154],"appears":[155],"be":[157],"Euclidean":[162],"case.":[163],"The":[164],"involve":[166],"standard":[168],"distribution":[170],"function":[171],"$\u03a6(x)$":[172],"its":[174],"values":[175],"purely":[177],"imaginary":[178],"arguments":[179],"$\u03a6(\\mathrm{i}":[180],"x)$.":[181],"example,":[183],"if":[184],"$d\\geq":[185],"2$":[186],"is":[187,205],"even,":[188],"then":[189],"ideal":[194],"simplex":[196],"$d$-dimensional":[199],"space":[201],"$\u03ba=-1$":[204],"given":[206],"by":[207],"absolutely":[209],"convergent":[210],"$$":[212,227],"\\frac{\u03c0^{d/2}}":[213],"{\\sqrt{2}\\,\\mathrm{i}^{d}\\,":[214],"\u0393\\left(\\frac{d+1}{2}\\right)}":[215],"\\int_{-\\infty}^{\\infty}":[216],"\\left[":[217],"\u03a6\\left(\\frac{\\mathrm{i}":[218],"y}{\\sqrt":[219,223],"d}\\right)^{d+1}":[220,224],"+":[221],"\u03a6\\left(-\\frac{\\mathrm{i}":[222],"\\right]":[225],"\\mathrm{e}^{-y^2/2}\\,\\mathrm{d}y.":[226]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-22T00:00:00"}