{"id":"https://openalex.org/W7204889372","doi":"https://doi.org/10.48550/arxiv.2608.30876","title":"Symmetry-dependence in Rounding of a Convex Body","display_name":"Symmetry-dependence in Rounding of a Convex Body","publication_year":2026,"publication_date":"2026-08-31","ids":{"openalex":"https://openalex.org/W7204889372","doi":"https://doi.org/10.48550/arxiv.2608.30876"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.30876","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30876","pdf_url":"https://arxiv.org/pdf/2608.30876","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/2608.30876","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5090740525","display_name":"Zikai Xiong","orcid":"https://orcid.org/0000-0003-3025-7846","observed_orcids":["https://orcid.org/0000-0003-3025-7846"]},"institutions":[{"id":"https://openalex.org/I111979921","display_name":"Northwestern University","ror":"https://ror.org/000e0be47","country_code":"US","type":"education","lineage":["https://openalex.org/I111979921"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xiong, Zikai","raw_affiliation_strings":["Department of Industrial Engineering and Management Sciences , Northwestern University , 2145 Sheridan Road , Evanston , IL 60208 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Industrial Engineering and Management Sciences , Northwestern University , 2145 Sheridan Road , Evanston , IL 60208 , USA","institution_ids":["https://openalex.org/I111979921"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101429270","display_name":"Robert M. Freund","orcid":"https://orcid.org/0000-0002-1733-5363","observed_orcids":["https://orcid.org/0000-0002-1733-5363"]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Freund, Robert M.","raw_affiliation_strings":["MIT Sloan School of Management , 77 Massachusetts Avenue , Cambridge , MA 02139 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"MIT Sloan School of Management , 77 Massachusetts Avenue , Cambridge , MA 02139 , USA","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"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/T11830","display_name":"Point processes and geometric inequalities","score":0.8521000146865845,"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":0.8521000146865845,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.08910000324249268,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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":0.019600000232458115,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/convex-bodies","display_name":"convex bodies","score":0.6150000095367432},{"id":"https://openalex.org/keywords/symmetry-measures","display_name":"symmetry measures","score":0.6759999990463257},{"id":"https://openalex.org/keywords/polytopes","display_name":"polytopes","score":0.7039999961853027},{"id":"https://openalex.org/keywords/semidefinite-programming","display_name":"semidefinite programming","score":0.625},{"id":"https://openalex.org/keywords/second-order-cone-constraints","display_name":"second-order cone constraints","score":0.8859999775886536}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8436999917030334},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6978999972343445},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.6575999855995178},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.6474999785423279},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.6039999723434448},{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.5773000121116638},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.554099977016449},{"id":"https://openalex.org/C57489055","wikidata":"https://www.wikidata.org/wiki/Q190046","display_name":"Ellipsoid","level":2,"score":0.5041999816894531},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.47780001163482666},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4569999873638153},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4268999993801117},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.4185999929904938},{"id":"https://openalex.org/C79464548","wikidata":"https://www.wikidata.org/wiki/Q464794","display_name":"Minkowski space","level":2,"score":0.3968999981880188},{"id":"https://openalex.org/C111110010","wikidata":"https://www.wikidata.org/wiki/Q2627315","display_name":"Convex combination","level":4,"score":0.390500009059906},{"id":"https://openalex.org/C44074806","wikidata":"https://www.wikidata.org/wiki/Q1066790","display_name":"Inscribed figure","level":2,"score":0.3903000056743622},{"id":"https://openalex.org/C10233890","wikidata":"https://www.wikidata.org/wiki/Q1322294","display_name":"Minkowski addition","level":3,"score":0.3255000114440918},{"id":"https://openalex.org/C200661725","wikidata":"https://www.wikidata.org/wiki/Q289500","display_name":"Subderivative","level":4,"score":0.311599999666214},{"id":"https://openalex.org/C105239961","wikidata":"https://www.wikidata.org/wiki/Q160003","display_name":"Tetrahedron","level":2,"score":0.3070000112056732},{"id":"https://openalex.org/C44550424","wikidata":"https://www.wikidata.org/wiki/Q4424496","display_name":"Mixed volume","level":5,"score":0.2671000063419342},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.26420000195503235},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.26179999113082886},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.2587999999523163}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.30876","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30876","pdf_url":"https://arxiv.org/pdf/2608.30876","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.2608.30876","doi":"https://doi.org/10.48550/arxiv.2608.30876","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.30876","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:2608.30876","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30876","pdf_url":"https://arxiv.org/pdf/2608.30876","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204889372.pdf","grobid_xml":"https://content.openalex.org/works/W7204889372.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7171819312"],"related_works":["https://openalex.org/W2154699848","https://openalex.org/W2503197712","https://openalex.org/W7095709190","https://openalex.org/W7220654807","https://openalex.org/W2049969034","https://openalex.org/W2138663260","https://openalex.org/W7014838027","https://openalex.org/W7131889380","https://openalex.org/W3099130640","https://openalex.org/W2022861720"],"abstract_inverted_index":{"The":[0],"symmetry":[1,232],"measure":[2],"of":[3,39,73,95,129,145],"a":[4,26,37,49,78,88,103,123,132,137,156,164,170,175,236],"convex":[5,33,93,171],"body":[6,34],"$S\\subset\\mathbb{R}^n$":[7],"is":[8,24,87,100,112,122,134,194,206,218,244],"given":[9,90,125],"by:":[10],"$\\mathrm{sym}(S):=\\max\\{\u03b1\\ge0:\\text{":[11],"there":[12,42],"exists":[13,43],"}x\\in":[14],"S\\text{":[15],"such":[16,21,52,131],"that":[17,31,53,190,214],"}-\u03b1(S-x)\\subseteq":[18],"S-x\\}$,":[19],"where":[20,109],"an":[22,44,71,207],"$x$":[23],"called":[25],"Minkowski":[27,117,165],"center.":[28,118],"We":[29,187],"prove":[30],"every":[32,226,230,241],"$S$":[35,74,86,121,238],"admits":[36],"$\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}$-rounding":[38],"$S$,":[40],"namely,":[41],"origin-centered":[45],"ellipsoid":[46,107,141],"$E$":[47],"and":[48,64,77,163,167,182,202,229],"center":[50],"$c$":[51],"$E\\subseteq":[54],"S-c\\subseteq\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}\\,E$.":[55],"This":[56],"result":[57],"was":[58],"conjectured":[59],"in":[60,197],"2005":[61],"by":[62,102,136,153,211],"Belloni":[63],"Freund.":[65],"As":[66],"special":[67],"cases,":[68,147],"this":[69],"recovers":[70],"$n$-rounding":[72],"(since":[75],"$\\mathrm{sym}(S)\\ge\\frac{1}{n}$),":[76],"$\\sqrt{n}$-rounding":[79],"when":[80,85,120],"$\\mathrm{sym}(S)=1$.":[81],"In":[82,143,220],"the":[83,92,97,110,116,127,148,191,215,221],"case":[84],"polytope":[89,124,237],"as":[91,126],"hull":[94],"points,":[96],"desired":[98],"rounding":[99,133,149,242],"produced":[101,135],"regularized":[104,138],"minimum-volume":[105],"covering":[106],"problem":[108,159,173],"regularization":[111],"with":[113,174],"respect":[114],"to":[115],"Similarly,":[119],"intersection":[128],"halfspaces,":[130],"maximum-volume":[139],"inscribed":[140],"problem.":[142],"both":[144],"these":[146],"can":[150],"be":[151],"computed":[152],"first":[154],"solving":[155,169],"linear":[157],"optimization":[158,172],"(to":[160],"compute":[161],"$\\mathrm{sym}(S)$":[162],"center),":[166],"then":[168],"logarithmic":[176],"determinant":[177],"objective,":[178],"second-order":[179],"cone":[180,185],"constraints,":[181],"one":[183],"semidefinite":[184],"constraint.":[186],"also":[188],"show":[189,210],"factor":[192,216,243],"$\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}$":[193,217],"nearly":[195],"tight":[196],"its":[198],"dependence":[199],"on":[200],"dimension":[201,227],"symmetry.":[203],"When":[204],"$\\frac{n+1}{1+\\mathrm{sym}(S)}$":[205],"integer,":[208],"we":[209,234],"explicit":[212],"construction":[213],"tight.":[219],"more":[222],"general":[223],"case,":[224],"for":[225,239],"$n$":[228],"admissible":[231],"value,":[233],"construct":[235],"which":[240],"at":[245],"least":[246],"$\\sqrt{\\frac{2}{3}}\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}$.":[247]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-02T00:00:00"}