{"id":"https://openalex.org/W7214029207","doi":"https://doi.org/10.48550/arxiv.2609.23913","title":"Fractional illumination and the optimal exponential rate in Hadwiger's covering conjecture","display_name":"Fractional illumination and the optimal exponential rate in Hadwiger's covering conjecture","publication_year":2026,"publication_date":"2026-09-20","ids":{"openalex":"https://openalex.org/W7214029207","doi":"https://doi.org/10.48550/arxiv.2609.23913"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.23913","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23913","pdf_url":"https://arxiv.org/pdf/2609.23913","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/2609.23913","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5152462381","display_name":"Yegor Gorodzha","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Gorodzha, Yegor","raw_affiliation_strings":["ETH Zurich , Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ETH Zurich , Switzerland","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5152462381"],"corresponding_institution_ids":["https://openalex.org/I35440088"],"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":0.9955999851226807,"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":0.9955999851226807,"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":0.002199999988079071,"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":0.0012000000569969416,"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/convex-bodies","display_name":"convex bodies","score":0.9210000038146973},{"id":"https://openalex.org/keywords/homothetic-copies","display_name":"homothetic copies","score":0.8939999938011169},{"id":"https://openalex.org/keywords/parallelotopes","display_name":"parallelotopes","score":0.8130000233650208}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.804099977016449},{"id":"https://openalex.org/C91709909","wikidata":"https://www.wikidata.org/wiki/Q583960","display_name":"Homothetic transformation","level":2,"score":0.6539000272750854},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6183000206947327},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5995000004768372},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5533999800682068},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5328999757766724},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5023000240325928},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.4900999963283539},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4731000065803528},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35580000281333923},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.35409998893737793},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.334199994802475},{"id":"https://openalex.org/C180188523","wikidata":"https://www.wikidata.org/wiki/Q574576","display_name":"Exponential decay","level":2,"score":0.3061000108718872},{"id":"https://openalex.org/C98184364","wikidata":"https://www.wikidata.org/wiki/Q1780131","display_name":"Argument (complex analysis)","level":2,"score":0.3027999997138977},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.29580000042915344},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28130000829696655},{"id":"https://openalex.org/C51823790","wikidata":"https://www.wikidata.org/wiki/Q504353","display_name":"Greedy algorithm","level":2,"score":0.27649998664855957},{"id":"https://openalex.org/C55350006","wikidata":"https://www.wikidata.org/wiki/Q237193","display_name":"Exponential distribution","level":2,"score":0.2694999873638153},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.2572999894618988},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.2565000057220459},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.2515999972820282},{"id":"https://openalex.org/C55974624","wikidata":"https://www.wikidata.org/wiki/Q1188504","display_name":"Exponential family","level":2,"score":0.2506999969482422}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.23913","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23913","pdf_url":"https://arxiv.org/pdf/2609.23913","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.2609.23913","doi":"https://doi.org/10.48550/arxiv.2609.23913","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.23913","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:2609.23913","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23913","pdf_url":"https://arxiv.org/pdf/2609.23913","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":[{"id":"https://openalex.org/F4320315473","display_name":"Institute of Science and Technology Austria","ror":"https://ror.org/03gnh5541"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214029207.pdf","grobid_xml":"https://content.openalex.org/works/W7214029207.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1187808733","https://openalex.org/W1596900230","https://openalex.org/W1967620596","https://openalex.org/W2002643080","https://openalex.org/W2011877429","https://openalex.org/W2011948965","https://openalex.org/W2038406532","https://openalex.org/W2052333181","https://openalex.org/W2055666178","https://openalex.org/W2163322487","https://openalex.org/W2902340877","https://openalex.org/W2964310552","https://openalex.org/W4386582663","https://openalex.org/W4413186473","https://openalex.org/W4413888283"],"related_works":["https://openalex.org/W7214859795","https://openalex.org/W2945540773","https://openalex.org/W2951620697","https://openalex.org/W170630596","https://openalex.org/W2020353244","https://openalex.org/W3088569456","https://openalex.org/W4400596755","https://openalex.org/W136524393","https://openalex.org/W2245249905","https://openalex.org/W2963993467"],"abstract_inverted_index":{"We":[0,22],"show":[1],"that":[2,25],"the":[3,45],"fractional":[4],"illumination":[5],"number":[6],"of":[7],"every":[8,26],"convex":[9],"body":[10,28],"in":[11,49],"$\\mathbb{R}^d$":[12],"is":[13],"at":[14,33],"most":[15,34],"$2^d$,":[16],"with":[17],"equality":[18],"exactly":[19],"for":[20],"parallelotopes.":[21],"also":[23],"prove":[24],"such":[27],"can":[29],"be":[30],"covered":[31],"by":[32,60],"$2^d(d\\log":[35],"d+d\\log\\log":[36],"d+O(d))$":[37],"smaller":[38],"positive":[39],"homothetic":[40],"copies":[41],"as":[42],"$d\\to\\infty$,":[43],"establishing":[44],"optimal":[46],"exponential":[47],"rate":[48],"Hadwiger's":[50],"covering":[51,57,68],"conjecture.":[52],"The":[53],"proofs":[54],"use":[55],"a":[56,66,71],"measure":[58],"obtained":[59],"minimizing":[61],"an":[62],"overlap":[63],"energy":[64],"and":[65],"greedy":[67],"argument":[69],"on":[70],"finite":[72],"net.":[73]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-23T00:00:00"}