{"publication_identifier":{"issn":["0024-6093"],"eissn":["1469-2120"]},"type":"journal_article","acknowledgement":"M.N. was supported by the János Bolyai Scholarship of the Hungarian Academy of Sciences aswell as the National Research, Development and Innovation Fund (NRDI) grants K119670 andK131529, and the ÚNKP-22-5 New National Excellence Program of the Ministry for Innovationand Technology from the source of the NRDI as well as the ELTE TKP 2021-NKTA-62 fundingscheme","article_processing_charge":"Yes (via OA deal)","date_published":"2023-12-04T00:00:00Z","title":"Quantitative Steinitz theorem: A polynomial bound","quality_controlled":"1","oa_version":"Published Version","abstract":[{"lang":"eng","text":"The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set 𝑆⊂ℝ𝑑, then there are at most 2𝑑 points of 𝑆 whose convex hull contains the origin in the interior. Bárány, Katchalski,and Pach proved the following quantitative version of Steinitz’s theorem. Let 𝑄 be a convex polytope in ℝ𝑑 containing the standard Euclidean unit ball 𝐁𝑑. Then there exist at most 2𝑑 vertices of 𝑄 whose convex hull 𝑄′ satisfies 𝑟𝐁𝑑⊂𝑄′ with 𝑟⩾𝑑−2𝑑. They conjectured that 𝑟⩾𝑐𝑑−1∕2 holds with a universal constant 𝑐>0. We prove 𝑟⩾15𝑑2, the first polynomial lower bound on 𝑟. Furthermore, we show that 𝑟 is not greater than 2/√𝑑."}],"doi":"10.1112/blms.12965","main_file_link":[{"url":" https://doi.org/10.1112/blms.12965","open_access":"1"}],"publication":"Bulletin of the London Mathematical Society","citation":{"ama":"Ivanov G, Naszódi M. Quantitative Steinitz theorem: A polynomial bound. <i>Bulletin of the London Mathematical Society</i>. 2023. doi:<a href=\"https://doi.org/10.1112/blms.12965\">10.1112/blms.12965</a>","ista":"Ivanov G, Naszódi M. 2023. Quantitative Steinitz theorem: A polynomial bound. Bulletin of the London Mathematical Society.","apa":"Ivanov, G., &#38; Naszódi, M. (2023). Quantitative Steinitz theorem: A polynomial bound. <i>Bulletin of the London Mathematical Society</i>. London Mathematical Society. <a href=\"https://doi.org/10.1112/blms.12965\">https://doi.org/10.1112/blms.12965</a>","short":"G. Ivanov, M. Naszódi, Bulletin of the London Mathematical Society (2023).","chicago":"Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A Polynomial Bound.” <i>Bulletin of the London Mathematical Society</i>. London Mathematical Society, 2023. <a href=\"https://doi.org/10.1112/blms.12965\">https://doi.org/10.1112/blms.12965</a>.","ieee":"G. Ivanov and M. Naszódi, “Quantitative Steinitz theorem: A polynomial bound,” <i>Bulletin of the London Mathematical Society</i>. London Mathematical Society, 2023.","mla":"Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A Polynomial Bound.” <i>Bulletin of the London Mathematical Society</i>, London Mathematical Society, 2023, doi:<a href=\"https://doi.org/10.1112/blms.12965\">10.1112/blms.12965</a>."},"_id":"14660","external_id":{"arxiv":["2212.04308"]},"scopus_import":"1","publisher":"London Mathematical Society","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"date_created":"2023-12-10T23:00:58Z","month":"12","date_updated":"2023-12-11T10:03:54Z","status":"public","department":[{"_id":"UlWa"}],"year":"2023","author":[{"last_name":"Ivanov","full_name":"Ivanov, Grigory","first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E"},{"first_name":"Márton","last_name":"Naszódi","full_name":"Naszódi, Márton"}],"language":[{"iso":"eng"}],"publication_status":"epub_ahead","day":"04","article_type":"original"}