[{"oa_version":"Preprint","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2103.04122"]},"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.","intvolume":" 36","volume":36,"department":[{"_id":"UlWa"}],"_id":"11435","publication_identifier":{"issn":["08954801"]},"author":[{"last_name":"Ivanov","first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","full_name":"Ivanov, Grigory"},{"first_name":"Marton","full_name":"Naszodi, Marton","last_name":"Naszodi"}],"type":"journal_article","article_processing_charge":"No","scopus_import":"1","language":[{"iso":"eng"}],"title":"A quantitative Helly-type theorem: Containment in a homothet","publication_status":"published","article_type":"original","publisher":"SIAM","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"}],"citation":{"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.” *SIAM Journal on Discrete Mathematics*. SIAM, 2022. https://doi.org/10.1137/21M1403308.","ieee":"G. Ivanov and M. Naszodi, “A quantitative Helly-type theorem: Containment in a homothet,” *SIAM Journal on Discrete Mathematics*, vol. 36, no. 2. SIAM, pp. 951–957, 2022.","apa":"Ivanov, G., & Naszodi, M. (2022). A quantitative Helly-type theorem: Containment in a homothet. *SIAM Journal on Discrete Mathematics*. SIAM. https://doi.org/10.1137/21M1403308","ama":"Ivanov G, Naszodi M. A quantitative Helly-type theorem: Containment in a homothet. *SIAM Journal on Discrete Mathematics*. 2022;36(2):951-957. doi:10.1137/21M1403308","mla":"Ivanov, Grigory, and Marton Naszodi. “A Quantitative Helly-Type Theorem: Containment in a Homothet.” *SIAM Journal on Discrete Mathematics*, vol. 36, no. 2, SIAM, 2022, pp. 951–57, doi:10.1137/21M1403308.","short":"G. Ivanov, M. Naszodi, SIAM Journal on Discrete Mathematics 36 (2022) 951–957."},"year":"2022","issue":"2","status":"public","date_published":"2022-04-11T00:00:00Z","date_updated":"2022-06-07T07:21:35Z","publication":"SIAM Journal on Discrete Mathematics","doi":"10.1137/21M1403308","date_created":"2022-06-05T22:01:50Z","day":"11","page":"951-957","month":"04","oa":1,"main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2103.04122"}]}]