{"id":"https://openalex.org/W2771985859","doi":"https://doi.org/10.1007/s41468-019-00029-8","title":"The reach, metric distortion, geodesic convexity and the variation of tangent spaces","display_name":"The reach, metric distortion, geodesic convexity and the variation of tangent spaces","publication_year":2019,"publication_date":"2019-06-01","ids":{"openalex":"https://openalex.org/W2771985859","doi":"https://doi.org/10.1007/s41468-019-00029-8","mag":"2771985859","pmid":"https://pubmed.ncbi.nlm.nih.gov/31633006"},"language":"en","primary_location":{"id":"doi:10.1007/s41468-019-00029-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s41468-019-00029-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s41468-019-00029-8.pdf","source":{"id":"https://openalex.org/S4210173698","display_name":"Journal of Applied and Computational Topology","issn_l":"2367-1726","issn":["2367-1726","2367-1734"],"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":"Journal of Applied and Computational Topology","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s41468-019-00029-8.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5105370783","display_name":"Jean\u2010Daniel Boissonnat","orcid":null},"institutions":[{"id":"https://openalex.org/I4210106545","display_name":"Research Centre Inria Sophia Antipolis - M\u00e9diterran\u00e9e","ror":"https://ror.org/01nzkaw91","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283","https://openalex.org/I201841394","https://openalex.org/I4210106545"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Jean-Daniel Boissonnat","raw_affiliation_strings":["1Inria Sophia Antipolis M\u00e9diterran\u00e9e, Universit\u00e9 C\u00f4te d'Azur, 2004 Route des Lucioles, 06902 Valbonne, France","Understanding the Shape of Data","Inria Sophia Antipolis - M\u00e9diterran\u00e9e","COMUE Universit\u00e9 C\u00f4te d'Azur (2015-2019)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"1Inria Sophia Antipolis M\u00e9diterran\u00e9e, Universit\u00e9 C\u00f4te d'Azur, 2004 Route des Lucioles, 06902 Valbonne, France","institution_ids":[]},{"raw_affiliation_string":"Understanding the Shape of Data","institution_ids":[]},{"raw_affiliation_string":"Inria Sophia Antipolis - M\u00e9diterran\u00e9e","institution_ids":["https://openalex.org/I4210106545"]},{"raw_affiliation_string":"COMUE Universit\u00e9 C\u00f4te d'Azur (2015-2019)","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032344519","display_name":"Andr\u00e9 Lieutier","orcid":"https://orcid.org/0000-0001-9517-4641"},"institutions":[{"id":"https://openalex.org/I47431100","display_name":"Dassault Syst\u00e8mes (France)","ror":"https://ror.org/038sc5x72","country_code":"FR","type":"company","lineage":["https://openalex.org/I47431100"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Andr\u00e9 Lieutier","raw_affiliation_strings":["3Dassault Syst\u00e8mes, 53 Avenue de l'Europe, 13090 Aix-en-Provence, France","Dassault Syst\u00e8mes"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"3Dassault Syst\u00e8mes, 53 Avenue de l'Europe, 13090 Aix-en-Provence, France","institution_ids":[]},{"raw_affiliation_string":"Dassault Syst\u00e8mes","institution_ids":["https://openalex.org/I47431100"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5003460394","display_name":"Mathijs Wintraecken","orcid":"https://orcid.org/0000-0002-7472-2220"},"institutions":[{"id":"https://openalex.org/I4210106545","display_name":"Research Centre Inria Sophia Antipolis - M\u00e9diterran\u00e9e","ror":"https://ror.org/01nzkaw91","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283","https://openalex.org/I201841394","https://openalex.org/I4210106545"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Mathijs Wintraecken","raw_affiliation_strings":["1Inria Sophia Antipolis M\u00e9diterran\u00e9e, Universit\u00e9 C\u00f4te d'Azur, 2004 Route des Lucioles, 06902 Valbonne, France","2Institute of Science and Technology (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria","Understanding the Shape of Data","Inria Sophia Antipolis - M\u00e9diterran\u00e9e"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"1Inria Sophia Antipolis M\u00e9diterran\u00e9e, Universit\u00e9 C\u00f4te d'Azur, 2004 Route des Lucioles, 06902 Valbonne, France","institution_ids":[]},{"raw_affiliation_string":"2Institute of Science and Technology (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria","institution_ids":[]},{"raw_affiliation_string":"Understanding the Shape of Data","institution_ids":[]},{"raw_affiliation_string":"Inria Sophia Antipolis - M\u00e9diterran\u00e9e","institution_ids":["https://openalex.org/I4210106545"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5105370783"],"corresponding_institution_ids":["https://openalex.org/I4210106545"],"apc_list":{"value":2490,"currency":"EUR","value_usd":3090},"apc_paid":{"value":2490,"currency":"EUR","value_usd":3090},"fwci":5.0541,"has_fulltext":true,"cited_by_count":43,"citation_normalized_percentile":{"value":0.95633172,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"3","issue":"1-2","first_page":"29","last_page":"58"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.996999979019165,"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"}},"topics":[{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.996999979019165,"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"}},{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9923999905586243,"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/T11830","display_name":"Point processes and geometric inequalities","score":0.9868999719619751,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7123392820358276},{"id":"https://openalex.org/keywords/geodesic","display_name":"Geodesic","score":0.657933235168457},{"id":"https://openalex.org/keywords/intersection","display_name":"Intersection (aeronautics)","score":0.5542007088661194},{"id":"https://openalex.org/keywords/convexity","display_name":"Convexity","score":0.507707953453064},{"id":"https://openalex.org/keywords/tangent-space","display_name":"Tangent space","score":0.5014607906341553},{"id":"https://openalex.org/keywords/ball","display_name":"Ball (mathematics)","score":0.4320923984050751},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4271399974822998},{"id":"https://openalex.org/keywords/shortest-path-problem","display_name":"Shortest path problem","score":0.42196792364120483},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.41161292791366577},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3939204216003418},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.32690611481666565}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7123392820358276},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.657933235168457},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.5542007088661194},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.507707953453064},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.5014607906341553},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.4320923984050751},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4271399974822998},{"id":"https://openalex.org/C22590252","wikidata":"https://www.wikidata.org/wiki/Q1058754","display_name":"Shortest path problem","level":3,"score":0.42196792364120483},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.41161292791366577},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3939204216003418},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32690611481666565},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"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/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s41468-019-00029-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s41468-019-00029-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s41468-019-00029-8.pdf","source":{"id":"https://openalex.org/S4210173698","display_name":"Journal of Applied and Computational Topology","issn_l":"2367-1726","issn":["2367-1726","2367-1734"],"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":"Journal of Applied and Computational Topology","raw_type":"journal-article"},{"id":"pmid:31633006","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/31633006","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of applied and computational topology","raw_type":null},{"id":"pmh:oai:HAL:hal-02266258v1","is_oa":false,"landing_page_url":"https://inria.hal.science/hal-02266258","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Applied and Computational Topology, 2019, &#x27E8;10.1007/s41468-019-00029-8&#x27E9;","raw_type":"Journal articles"},{"id":"pmh:oai:pubmedcentral.nih.gov:6777504","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6777504","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"J Appl Comput Topol","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1007/s41468-019-00029-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s41468-019-00029-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s41468-019-00029-8.pdf","source":{"id":"https://openalex.org/S4210173698","display_name":"Journal of Applied and Computational Topology","issn_l":"2367-1726","issn":["2367-1726","2367-1734"],"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":"Journal of Applied and Computational Topology","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.6499999761581421,"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11"}],"awards":[{"id":"https://openalex.org/G2761285720","display_name":null,"funder_award_id":"FP/2007-2013","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G3216667337","display_name":null,"funder_award_id":"339025","funder_id":"https://openalex.org/F4320334678","funder_display_name":"European Research Council"},{"id":"https://openalex.org/G3738360915","display_name":null,"funder_award_id":"Grant Agreement No. 754411","funder_id":"https://openalex.org/F4320315473","funder_display_name":"Institute of Science and Technology Austria"},{"id":"https://openalex.org/G4101941094","display_name":"ISTplus - Postdoctoral Fellowships","funder_award_id":"754411","funder_id":"https://openalex.org/F4320338337","funder_display_name":"H2020 Marie Sk\u0142odowska-Curie Actions"},{"id":"https://openalex.org/G4470488369","display_name":null,"funder_award_id":"754411","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G5593277320","display_name":null,"funder_award_id":"2007-2013","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G6309856603","display_name":"Algorithmic Foundations of Geometry Understanding in Higher Dimensions","funder_award_id":"339025","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G7546946110","display_name":null,"funder_award_id":"FP/2007-2013)/ERC","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8049466214","display_name":null,"funder_award_id":"754411","funder_id":"https://openalex.org/F4320315473","funder_display_name":"Institute of Science and Technology Austria"}],"funders":[{"id":"https://openalex.org/F4320315473","display_name":"Institute of Science and Technology Austria","ror":"https://ror.org/03gnh5541"},{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320334678","display_name":"European Research Council","ror":"https://ror.org/0472cxd90"},{"id":"https://openalex.org/F4320338337","display_name":"H2020 Marie Sk\u0142odowska-Curie Actions","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2771985859.pdf","grobid_xml":"https://content.openalex.org/works/W2771985859.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W156852151","https://openalex.org/W1483679902","https://openalex.org/W1507196244","https://openalex.org/W1714207109","https://openalex.org/W1921997421","https://openalex.org/W1922506815","https://openalex.org/W1988523136","https://openalex.org/W1989057772","https://openalex.org/W2003907241","https://openalex.org/W2006063729","https://openalex.org/W2029041800","https://openalex.org/W2039934664","https://openalex.org/W2064311953","https://openalex.org/W2067265108","https://openalex.org/W2083493892","https://openalex.org/W2099963367","https://openalex.org/W2122950280","https://openalex.org/W2157878484","https://openalex.org/W2319530283","https://openalex.org/W2794362239","https://openalex.org/W3038420293","https://openalex.org/W3101870792","https://openalex.org/W4206033904","https://openalex.org/W4213171111","https://openalex.org/W4235974481","https://openalex.org/W4238357952","https://openalex.org/W4249413533","https://openalex.org/W4310845630"],"related_works":["https://openalex.org/W1964872188","https://openalex.org/W2187391117","https://openalex.org/W1981809986","https://openalex.org/W2949367465","https://openalex.org/W2587048717","https://openalex.org/W4235203617","https://openalex.org/W2006359264","https://openalex.org/W3029201158","https://openalex.org/W1741548419","https://openalex.org/W4317607601"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,16,53,94],"discuss":[4],"three":[5],"results.":[6],"The":[7],"first":[8,17],"two":[9,80,112,125],"concern":[10],"general":[11],"sets":[12],"of":[13,21,27,58,117],"positive":[14,99],"reach:":[15],"characterize":[18],"the":[19,31,35,39,44,50,56,65,68,75,83,88,106,118,121,124],"reach":[20,66,100,119],"a":[22,28,59,103],"closed":[23],"set":[24,69],"by":[25],"means":[26],"bound":[29,104],"on":[30,96,105],"metric":[32],"distortion":[33],"between":[34,78,108,123],"distance":[36,47,122],"measured":[37,48],"in":[38,49,82,87,115],"ambient":[40],"Euclidean":[41],"space":[42],"and":[43,101,120],"shortest":[45,76],"path":[46,77],"set.":[51],"Secondly,":[52],"prove":[54],"that":[55,74],"intersection":[57,84],"ball":[60],"with":[61,67,98],"radius":[62],"less":[63],"than":[64],"is":[70],"geodesically":[71],"convex,":[72],"meaning":[73],"any":[79],"points":[81,114],"lies":[85],"itself":[86],"intersection.":[89],"For":[90],"our":[91],"third":[92],"result":[93],"focus":[95],"manifolds":[97],"give":[102],"angle":[107],"tangent":[109],"spaces":[110],"at":[111],"different":[113],"terms":[116],"points.":[126]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":2}],"updated_date":"2026-05-17T08:19:37.847499","created_date":"2025-10-10T00:00:00"}
