{"id":"https://openalex.org/W2020167116","doi":"https://doi.org/10.1137/0804006","title":"Existence and Differentiability of Metric Projections in Hilbert Spaces","display_name":"Existence and Differentiability of Metric Projections in Hilbert Spaces","publication_year":1994,"publication_date":"1994-02-01","ids":{"openalex":"https://openalex.org/W2020167116","doi":"https://doi.org/10.1137/0804006","mag":"2020167116"},"language":"en","primary_location":{"id":"doi:10.1137/0804006","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0804006","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-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/A5068780452","display_name":"Alexander Shapiro","orcid":"https://orcid.org/0000-0002-4507-0534"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Alexander Shapiro","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5068780452"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.53851926,"has_fulltext":false,"cited_by_count":71,"citation_normalized_percentile":{"value":0.55879791,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"4","issue":"1","first_page":"130","last_page":"141"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9986000061035156,"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/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9986000061035156,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.9975000023841858,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.9509000182151794,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8711632490158081},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.7801480293273926},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.614734411239624},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.6085061430931091},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5376317501068115},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.5185721516609192},{"id":"https://openalex.org/keywords/intrinsic-metric","display_name":"Intrinsic metric","score":0.5020980834960938},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.49755099415779114},{"id":"https://openalex.org/keywords/convex-metric-space","display_name":"Convex metric space","score":0.4971802532672882},{"id":"https://openalex.org/keywords/fisher-information-metric","display_name":"Fisher information metric","score":0.49306246638298035},{"id":"https://openalex.org/keywords/injective-metric-space","display_name":"Injective metric space","score":0.4859975278377533},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.44901391863822937},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.425975501537323},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.39576756954193115},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.21932199597358704},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10177576541900635}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8711632490158081},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.7801480293273926},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.614734411239624},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.6085061430931091},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5376317501068115},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.5185721516609192},{"id":"https://openalex.org/C41261874","wikidata":"https://www.wikidata.org/wiki/Q3855712","display_name":"Intrinsic metric","level":4,"score":0.5020980834960938},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.49755099415779114},{"id":"https://openalex.org/C196613150","wikidata":"https://www.wikidata.org/wiki/Q3966092","display_name":"Convex metric space","level":3,"score":0.4971802532672882},{"id":"https://openalex.org/C127104897","wikidata":"https://www.wikidata.org/wiki/Q5454858","display_name":"Fisher information metric","level":4,"score":0.49306246638298035},{"id":"https://openalex.org/C177547787","wikidata":"https://www.wikidata.org/wiki/Q6034319","display_name":"Injective metric space","level":3,"score":0.4859975278377533},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.44901391863822937},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.425975501537323},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39576756954193115},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.21932199597358704},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10177576541900635},{"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:10.1137/0804006","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0804006","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.5199999809265137}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W30959786","https://openalex.org/W135273259","https://openalex.org/W135911866","https://openalex.org/W1968850685","https://openalex.org/W1969748223","https://openalex.org/W1976462192","https://openalex.org/W1982632325","https://openalex.org/W1992639195","https://openalex.org/W2016663415","https://openalex.org/W2018508199","https://openalex.org/W2025874681","https://openalex.org/W2036613704","https://openalex.org/W2068254260","https://openalex.org/W2081600571","https://openalex.org/W2116598569","https://openalex.org/W2595644070"],"related_works":["https://openalex.org/W4363675004","https://openalex.org/W2947185879","https://openalex.org/W2987243377","https://openalex.org/W2048731283","https://openalex.org/W2398868469","https://openalex.org/W4255883801","https://openalex.org/W2001114394","https://openalex.org/W2095137273","https://openalex.org/W2039845181","https://openalex.org/W4387521419"],"abstract_inverted_index":{"This":[0],"paper":[1],"considers":[2],"metric":[3,28,44,60],"projections":[4,29,45,61],"onto":[5],"a":[6,11],"closed":[7],"subset":[8],"S":[9,17],"of":[10,39,43,59,65],"Hilbert":[12],"space.":[13],"If":[14],"the":[15,26],"set":[16],"is":[18,22],"convex,":[19],"then":[20],"it":[21],"well":[23],"known":[24],"that":[25],"corresponding":[27],"always":[30],"exist,":[31],"unique":[32],"and":[33,56,69],"directionally":[34],"differentiable":[35],"at":[36],"boundary":[37],"points":[38],"S.":[40,52],"These":[41],"properties":[42],"are":[46,67],"considered":[47],"for":[48,62],"possibly":[49],"nonconvex":[50],"sets":[51,66],"In":[53],"particular,":[54],"existence":[55],"directional":[57],"differentiability":[58],"certain":[63],"classes":[64],"established":[68],"will":[70],"be":[71],"referred":[72],"to":[73],"as":[74],"\"nearly":[75],"convex\"":[76],"sets.":[77]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}