{"page":"951-957","doi":"10.1137/21M1403308","article_type":"original","type":"journal_article","issue":"2","acknowledgement":"G.I. acknowledges the financial support from the Ministry of Educational and Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926. M.N. was supported by the National Research, Development and Innovation Fund (NRDI) grants K119670 and\r\nKKP-133864 as well as the Bolyai Scholarship of the Hungarian Academy of Sciences and the New National Excellence Programme and the TKP2020-NKA-06 program provided by the NRDI.","status":"public","external_id":{"isi":["000793158200002"],"arxiv":["2103.04122"]},"author":[{"id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","full_name":"Ivanov, Grigory","last_name":"Ivanov","first_name":"Grigory"},{"first_name":"Marton","last_name":"Naszodi","full_name":"Naszodi, Marton"}],"oa_version":"Preprint","date_updated":"2023-10-18T06:58:03Z","month":"04","year":"2022","publication_identifier":{"issn":["0895-4801"]},"title":"A quantitative Helly-type theorem: Containment in a homothet","quality_controlled":"1","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2103.04122","open_access":"1"}],"intvolume":"        36","citation":{"ama":"Ivanov G, Naszodi M. A quantitative Helly-type theorem: Containment in a homothet. <i>SIAM Journal on Discrete Mathematics</i>. 2022;36(2):951-957. doi:<a href=\"https://doi.org/10.1137/21M1403308\">10.1137/21M1403308</a>","ieee":"G. Ivanov and M. Naszodi, “A quantitative Helly-type theorem: Containment in a homothet,” <i>SIAM Journal on Discrete Mathematics</i>, vol. 36, no. 2. Society for Industrial and Applied Mathematics, pp. 951–957, 2022.","mla":"Ivanov, Grigory, and Marton Naszodi. “A Quantitative Helly-Type Theorem: Containment in a Homothet.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 36, no. 2, Society for Industrial and Applied Mathematics, 2022, pp. 951–57, doi:<a href=\"https://doi.org/10.1137/21M1403308\">10.1137/21M1403308</a>.","short":"G. Ivanov, M. Naszodi, SIAM Journal on Discrete Mathematics 36 (2022) 951–957.","apa":"Ivanov, G., &#38; Naszodi, M. (2022). A quantitative Helly-type theorem: Containment in a homothet. <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/21M1403308\">https://doi.org/10.1137/21M1403308</a>","ista":"Ivanov G, Naszodi M. 2022. A quantitative Helly-type theorem: Containment in a homothet. SIAM Journal on Discrete Mathematics. 36(2), 951–957.","chicago":"Ivanov, Grigory, and Marton Naszodi. “A Quantitative Helly-Type Theorem: Containment in a Homothet.” <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics, 2022. <a href=\"https://doi.org/10.1137/21M1403308\">https://doi.org/10.1137/21M1403308</a>."},"date_published":"2022-04-11T00:00:00Z","_id":"11435","scopus_import":"1","oa":1,"publication_status":"published","language":[{"iso":"eng"}],"publication":"SIAM Journal on Discrete Mathematics","date_created":"2022-06-05T22:01:50Z","abstract":[{"text":"We introduce a new variant of quantitative Helly-type theorems: the minimal homothetic distance of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the diameter. If $K$ is the intersection of finitely many convex bodies in $\\mathbb{R}^d$, then one can select $2d$ of these bodies whose intersection is of diameter at most $(2d)^3{diam}(K)$. The best previously known estimate, due to Brazitikos [Bull. Hellenic Math. Soc., 62 (2018), pp. 19--25], is $c d^{11/2}$. Moreover, we confirm that the multiplicative factor $c d^{1/2}$ conjectured by Bárány, Katchalski, and Pach [Proc. Amer. Math. Soc., 86 (1982), pp. 109--114] cannot be improved. The bounds above follow from our key result that concerns sparse approximation of a convex polytope by the convex hull of a well-chosen subset of its vertices: Assume that $Q \\subset {\\mathbb R}^d$ is a polytope whose centroid is the origin. Then there exist at most 2d vertices of $Q$ whose convex hull $Q^{\\prime \\prime}$ satisfies $Q \\subset - 8d^3 Q^{\\prime \\prime}.$","lang":"eng"}],"volume":36,"isi":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","day":"11","publisher":"Society for Industrial and Applied Mathematics","department":[{"_id":"UlWa"}]}