[{"publication":"Discrete Mathematics","day":"01","year":"2023","date_created":"2023-02-26T23:01:00Z","doi":"10.1016/j.disc.2023.113363","date_published":"2023-06-01T00:00:00Z","oa":1,"publisher":"Elsevier","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Ivanov, Grigory, and Seyda Köse. “Erdős-Ko-Rado and Hilton-Milner Theorems for Two-Forms.” Discrete Mathematics, vol. 346, no. 6, 113363, Elsevier, 2023, doi:10.1016/j.disc.2023.113363.","apa":"Ivanov, G., & Köse, S. (2023). Erdős-Ko-Rado and Hilton-Milner theorems for two-forms. Discrete Mathematics. Elsevier. https://doi.org/10.1016/j.disc.2023.113363","ama":"Ivanov G, Köse S. Erdős-Ko-Rado and Hilton-Milner theorems for two-forms. Discrete Mathematics. 2023;346(6). doi:10.1016/j.disc.2023.113363","ieee":"G. Ivanov and S. Köse, “Erdős-Ko-Rado and Hilton-Milner theorems for two-forms,” Discrete Mathematics, vol. 346, no. 6. Elsevier, 2023.","short":"G. Ivanov, S. Köse, Discrete Mathematics 346 (2023).","chicago":"Ivanov, Grigory, and Seyda Köse. “Erdős-Ko-Rado and Hilton-Milner Theorems for Two-Forms.” Discrete Mathematics. Elsevier, 2023. https://doi.org/10.1016/j.disc.2023.113363.","ista":"Ivanov G, Köse S. 2023. Erdős-Ko-Rado and Hilton-Milner theorems for two-forms. Discrete Mathematics. 346(6), 113363."},"title":"Erdős-Ko-Rado and Hilton-Milner theorems for two-forms","external_id":{"arxiv":["2201.10892"]},"article_processing_charge":"No","author":[{"first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","last_name":"Ivanov","full_name":"Ivanov, Grigory"},{"id":"8ba3170d-dc85-11ea-9058-c4251c96a6eb","first_name":"Seyda","full_name":"Köse, Seyda","last_name":"Köse"}],"article_number":"113363","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0012-365X"]},"related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"13331"}]},"issue":"6","volume":346,"oa_version":"Preprint","abstract":[{"text":"The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting family of r-element subsets of was extended to the setting of exterior algebra in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado theorem and the characterization of the equality case therein, as well as those of the Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from a folklore puzzle about possible arrangements of an intersecting family of lines.","lang":"eng"}],"intvolume":" 346","month":"06","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2201.10892","open_access":"1"}],"scopus_import":"1","date_updated":"2023-10-04T11:54:57Z","department":[{"_id":"UlWa"},{"_id":"GradSch"}],"_id":"12680","status":"public","article_type":"letter_note","type":"journal_article"},{"status":"public","article_type":"original","type":"journal_article","_id":"14660","title":"Quantitative Steinitz theorem: A polynomial bound","department":[{"_id":"UlWa"}],"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"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"arxiv":["2212.04308"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2023-12-11T10:03:54Z","citation":{"ama":"Ivanov G, Naszódi M. Quantitative Steinitz theorem: A polynomial bound. Bulletin of the London Mathematical Society. 2023. doi:10.1112/blms.12965","apa":"Ivanov, G., & Naszódi, M. (2023). Quantitative Steinitz theorem: A polynomial bound. Bulletin of the London Mathematical Society. London Mathematical Society. https://doi.org/10.1112/blms.12965","short":"G. Ivanov, M. Naszódi, Bulletin of the London Mathematical Society (2023).","ieee":"G. Ivanov and M. Naszódi, “Quantitative Steinitz theorem: A polynomial bound,” Bulletin of the London Mathematical Society. London Mathematical Society, 2023.","mla":"Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A Polynomial Bound.” Bulletin of the London Mathematical Society, London Mathematical Society, 2023, doi:10.1112/blms.12965.","ista":"Ivanov G, Naszódi M. 2023. Quantitative Steinitz theorem: A polynomial bound. Bulletin of the London Mathematical Society.","chicago":"Ivanov, Grigory, and Márton Naszódi. “Quantitative Steinitz Theorem: A Polynomial Bound.” Bulletin of the London Mathematical Society. London Mathematical Society, 2023. https://doi.org/10.1112/blms.12965."},"month":"12","scopus_import":"1","quality_controlled":"1","publisher":"London Mathematical Society","main_file_link":[{"url":" https://doi.org/10.1112/blms.12965","open_access":"1"}],"oa":1,"oa_version":"Published Version","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","abstract":[{"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.","lang":"eng"}],"doi":"10.1112/blms.12965","date_published":"2023-12-04T00:00:00Z","date_created":"2023-12-10T23:00:58Z","day":"04","language":[{"iso":"eng"}],"publication":"Bulletin of the London Mathematical Society","publication_identifier":{"issn":["0024-6093"],"eissn":["1469-2120"]},"publication_status":"epub_ahead","year":"2023"},{"ddc":["510"],"date_updated":"2024-01-08T09:57:25Z","department":[{"_id":"UlWa"}],"file_date_updated":"2024-01-08T09:53:09Z","_id":"14737","status":"public","keyword":["General Mathematics"],"article_type":"original","type":"journal_article","tmp":{"short":"CC BY-NC-ND (4.0)","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","image":"/images/cc_by_nc_nd.png"},"file":[{"date_created":"2024-01-08T09:53:09Z","file_name":"2023_IMRN_Ivanov.pdf","creator":"dernst","date_updated":"2024-01-08T09:53:09Z","file_size":815777,"file_id":"14738","checksum":"353666cea80633beb0f1ffd342dff6d4","success":1,"access_level":"open_access","relation":"main_file","content_type":"application/pdf"}],"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"publication_status":"published","volume":2023,"issue":"23","license":"https://creativecommons.org/licenses/by-nc-nd/4.0/","oa_version":"Published Version","abstract":[{"lang":"eng","text":"John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\\mathbb{R}^{d}$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called Löwner’s ellipsoid, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the Löwner problems is more intricate than it is in the setting of convex bodies."}],"month":"12","intvolume":" 2023","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"chicago":"Ivanov, Grigory, and Márton Naszódi. “Functional John and Löwner Conditions for Pairs of Log-Concave Functions.” International Mathematics Research Notices. Oxford University Press, 2023. https://doi.org/10.1093/imrn/rnad210.","ista":"Ivanov G, Naszódi M. 2023. Functional John and Löwner conditions for pairs of log-concave functions. International Mathematics Research Notices. 2023(23), 20613–20669.","mla":"Ivanov, Grigory, and Márton Naszódi. “Functional John and Löwner Conditions for Pairs of Log-Concave Functions.” International Mathematics Research Notices, vol. 2023, no. 23, Oxford University Press, 2023, pp. 20613–69, doi:10.1093/imrn/rnad210.","ieee":"G. Ivanov and M. Naszódi, “Functional John and Löwner conditions for pairs of log-concave functions,” International Mathematics Research Notices, vol. 2023, no. 23. Oxford University Press, pp. 20613–20669, 2023.","short":"G. Ivanov, M. Naszódi, International Mathematics Research Notices 2023 (2023) 20613–20669.","ama":"Ivanov G, Naszódi M. Functional John and Löwner conditions for pairs of log-concave functions. International Mathematics Research Notices. 2023;2023(23):20613-20669. doi:10.1093/imrn/rnad210","apa":"Ivanov, G., & Naszódi, M. (2023). Functional John and Löwner conditions for pairs of log-concave functions. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnad210"},"title":"Functional John and Löwner conditions for pairs of log-concave functions","author":[{"first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","full_name":"Ivanov, Grigory","last_name":"Ivanov"},{"first_name":"Márton","full_name":"Naszódi, Márton","last_name":"Naszódi"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"arxiv":["2212.11781"]},"day":"01","publication":"International Mathematics Research Notices","has_accepted_license":"1","year":"2023","doi":"10.1093/imrn/rnad210","date_published":"2023-12-01T00:00:00Z","date_created":"2024-01-08T09:48:56Z","page":"20613-20669","acknowledgement":"We thank Alexander Litvak for the many discussions on Theorem 1.1. Igor Tsiutsiurupa participated in the early stage of this project. To our deep regret, Igor chose another road for his life and stopped working with us.\r\nThis work was supported by the János Bolyai Scholarship of the Hungarian Academy of Sciences [to M.N.]; the National Research, Development, and Innovation Fund (NRDI) [K119670 and K131529 to M.N.]; and the ÚNKP-22-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the NRDI [to M.N.].","publisher":"Oxford University Press","quality_controlled":"1","oa":1},{"acknowledgement":"G.I. was supported by the Ministry of Education 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 KKP-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. ","oa":1,"publisher":"Elsevier","quality_controlled":"1","publication":"Journal of Functional Analysis","day":"01","year":"2022","isi":1,"has_accepted_license":"1","date_created":"2022-03-20T23:01:38Z","doi":"10.1016/j.jfa.2022.109441","date_published":"2022-06-01T00:00:00Z","article_number":"109441","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"short":"G. Ivanov, M. Naszódi, Journal of Functional Analysis 282 (2022).","ieee":"G. Ivanov and M. Naszódi, “Functional John ellipsoids,” Journal of Functional Analysis, vol. 282, no. 11. Elsevier, 2022.","apa":"Ivanov, G., & Naszódi, M. (2022). Functional John ellipsoids. Journal of Functional Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2022.109441","ama":"Ivanov G, Naszódi M. Functional John ellipsoids. Journal of Functional Analysis. 2022;282(11). doi:10.1016/j.jfa.2022.109441","mla":"Ivanov, Grigory, and Márton Naszódi. “Functional John Ellipsoids.” Journal of Functional Analysis, vol. 282, no. 11, 109441, Elsevier, 2022, doi:10.1016/j.jfa.2022.109441.","ista":"Ivanov G, Naszódi M. 2022. Functional John ellipsoids. Journal of Functional Analysis. 282(11), 109441.","chicago":"Ivanov, Grigory, and Márton Naszódi. “Functional John Ellipsoids.” Journal of Functional Analysis. Elsevier, 2022. https://doi.org/10.1016/j.jfa.2022.109441."},"title":"Functional John ellipsoids","external_id":{"arxiv":["2006.09934"],"isi":["000781371300008"]},"article_processing_charge":"Yes (via OA deal)","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"}],"oa_version":"Published Version","abstract":[{"lang":"eng","text":"We introduce a new way of representing logarithmically concave functions on Rd. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every s>0, we define a class of non-negative functions on Rd derived from ellipsoids in Rd+1. For any log-concave function f on Rd , and any fixed s>0, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to f, and we call it the John s-function of f. After establishing existence and uniqueness, we give a characterization of this function similar to the one given by John in his fundamental theorem. We find that John s-functions converge to characteristic functions of ellipsoids as s tends to zero and to Gaussian densities as s tends to infinity.\r\nAs an application, we prove a quantitative Helly type result: the integral of the pointwise minimum of any family of log-concave functions is at least a constant cd multiple of the integral of the pointwise minimum of a properly chosen subfamily of size 3d+2, where cd depends only on d."}],"intvolume":" 282","month":"06","scopus_import":"1","language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","access_level":"open_access","relation":"main_file","file_id":"11721","checksum":"1cf185e264e04c87cb1ce67a00db88ab","success":1,"date_updated":"2022-08-02T10:40:48Z","file_size":734482,"creator":"dernst","date_created":"2022-08-02T10:40:48Z","file_name":"2022_JourFunctionalAnalysis_Ivanov.pdf"}],"publication_status":"published","publication_identifier":{"issn":["0022-1236"],"eissn":["1096-0783"]},"license":"https://creativecommons.org/licenses/by/4.0/","issue":"11","volume":282,"_id":"10887","status":"public","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","ddc":["510"],"date_updated":"2023-08-02T14:51:11Z","file_date_updated":"2022-08-02T10:40:48Z","department":[{"_id":"UlWa"}]},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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. Society for Industrial and Applied Mathematics, 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. Society for Industrial and Applied Mathematics, pp. 951–957, 2022.","short":"G. Ivanov, M. Naszodi, SIAM Journal on Discrete Mathematics 36 (2022) 951–957.","apa":"Ivanov, G., & Naszodi, M. (2022). A quantitative Helly-type theorem: Containment in a homothet. SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics. 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, Society for Industrial and Applied Mathematics, 2022, pp. 951–57, doi:10.1137/21M1403308."},"title":"A quantitative Helly-type theorem: Containment in a homothet","author":[{"full_name":"Ivanov, Grigory","last_name":"Ivanov","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","first_name":"Grigory"},{"full_name":"Naszodi, Marton","last_name":"Naszodi","first_name":"Marton"}],"external_id":{"arxiv":["2103.04122"],"isi":["000793158200002"]},"article_processing_charge":"No","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.","publisher":"Society for Industrial and Applied Mathematics","quality_controlled":"1","oa":1,"day":"11","publication":"SIAM Journal on Discrete Mathematics","isi":1,"year":"2022","date_published":"2022-04-11T00:00:00Z","doi":"10.1137/21M1403308","date_created":"2022-06-05T22:01:50Z","page":"951-957","_id":"11435","status":"public","article_type":"original","type":"journal_article","date_updated":"2023-10-18T06:58:03Z","department":[{"_id":"UlWa"}],"oa_version":"Preprint","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"}],"month":"04","intvolume":" 36","scopus_import":"1","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2103.04122","open_access":"1"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["0895-4801"]},"publication_status":"published","issue":"2","volume":36},{"external_id":{"arxiv":["1912.08561"],"isi":["000607265100001"]},"article_processing_charge":"Yes (via OA deal)","author":[{"id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","first_name":"Grigory","full_name":"Ivanov, Grigory","last_name":"Ivanov"}],"title":"No-dimension Tverberg's theorem and its corollaries in Banach spaces of type p","citation":{"chicago":"Ivanov, Grigory. “No-Dimension Tverberg’s Theorem and Its Corollaries in Banach Spaces of Type P.” Bulletin of the London Mathematical Society. London Mathematical Society, 2021. https://doi.org/10.1112/blms.12449.","ista":"Ivanov G. 2021. No-dimension Tverberg’s theorem and its corollaries in Banach spaces of type p. Bulletin of the London Mathematical Society. 53(2), 631–641.","mla":"Ivanov, Grigory. “No-Dimension Tverberg’s Theorem and Its Corollaries in Banach Spaces of Type P.” Bulletin of the London Mathematical Society, vol. 53, no. 2, London Mathematical Society, 2021, pp. 631–41, doi:10.1112/blms.12449.","short":"G. Ivanov, Bulletin of the London Mathematical Society 53 (2021) 631–641.","ieee":"G. Ivanov, “No-dimension Tverberg’s theorem and its corollaries in Banach spaces of type p,” Bulletin of the London Mathematical Society, vol. 53, no. 2. London Mathematical Society, pp. 631–641, 2021.","apa":"Ivanov, G. (2021). No-dimension Tverberg’s theorem and its corollaries in Banach spaces of type p. Bulletin of the London Mathematical Society. London Mathematical Society. https://doi.org/10.1112/blms.12449","ama":"Ivanov G. No-dimension Tverberg’s theorem and its corollaries in Banach spaces of type p. Bulletin of the London Mathematical Society. 2021;53(2):631-641. doi:10.1112/blms.12449"},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"quality_controlled":"1","publisher":"London Mathematical Society","acknowledgement":"I wish to thank Imre Bárány for bringing the problem to my attention. I am grateful to Marton Naszódi and Igor Tsiutsiurupa for useful remarks and help with the text.\r\nThe author 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.","page":"631-641","date_created":"2021-01-24T23:01:08Z","doi":"10.1112/blms.12449","date_published":"2021-04-01T00:00:00Z","year":"2021","isi":1,"has_accepted_license":"1","publication":"Bulletin of the London Mathematical Society","day":"01","tmp":{"short":"CC BY-NC-ND (4.0)","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","image":"/images/cc_by_nc_nd.png"},"article_type":"original","type":"journal_article","status":"public","_id":"9037","file_date_updated":"2021-08-06T09:59:45Z","department":[{"_id":"UlWa"}],"date_updated":"2023-08-07T13:35:20Z","ddc":["510"],"scopus_import":"1","intvolume":" 53","month":"04","abstract":[{"text":"We continue our study of ‘no‐dimension’ analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper (Adiprasito, Bárány and Mustafa, ‘Theorems of Carathéodory, Helly, and Tverberg without dimension’, Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego, California, 2019) 2350–2360) and prove no‐dimension versions of the colored Tverberg theorem, the selection lemma and the weak 𝜀 ‐net theorem in Banach spaces of type 𝑝>1 . To prove these results, we use the original ideas of Adiprasito, Bárány and Mustafa for the Euclidean case, our no‐dimension version of the Radon theorem and slightly modified version of the celebrated Maurey lemma.","lang":"eng"}],"oa_version":"Published Version","volume":53,"issue":"2","publication_status":"published","publication_identifier":{"issn":["00246093"],"eissn":["14692120"]},"language":[{"iso":"eng"}],"file":[{"date_created":"2021-08-06T09:59:45Z","file_name":"2021_BLMS_Ivanov.pdf","date_updated":"2021-08-06T09:59:45Z","file_size":194550,"creator":"kschuh","file_id":"9796","checksum":"e6ceaa6470d835eb4c211cbdd38fdfd1","success":1,"content_type":"application/pdf","access_level":"open_access","relation":"main_file"}]},{"acknowledgement":"Research was supported by the Russian Foundation for Basic Research, project 18-01-00036A (Theorems 1.5 and 5.3) and by the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926 (Theorems 1.2 and 7.3).","oa":1,"publisher":"Elsevier","quality_controlled":"1","year":"2021","isi":1,"publication":"Discrete Mathematics","day":"01","date_created":"2021-02-07T23:01:12Z","doi":"10.1016/j.disc.2021.112312","date_published":"2021-05-01T00:00:00Z","article_number":"112312","citation":{"chicago":"Ivanov, Grigory. “On the Volume of Projections of the Cross-Polytope.” Discrete Mathematics. Elsevier, 2021. https://doi.org/10.1016/j.disc.2021.112312.","ista":"Ivanov G. 2021. On the volume of projections of the cross-polytope. Discrete Mathematics. 344(5), 112312.","mla":"Ivanov, Grigory. “On the Volume of Projections of the Cross-Polytope.” Discrete Mathematics, vol. 344, no. 5, 112312, Elsevier, 2021, doi:10.1016/j.disc.2021.112312.","short":"G. Ivanov, Discrete Mathematics 344 (2021).","ieee":"G. Ivanov, “On the volume of projections of the cross-polytope,” Discrete Mathematics, vol. 344, no. 5. Elsevier, 2021.","apa":"Ivanov, G. (2021). On the volume of projections of the cross-polytope. Discrete Mathematics. Elsevier. https://doi.org/10.1016/j.disc.2021.112312","ama":"Ivanov G. On the volume of projections of the cross-polytope. Discrete Mathematics. 2021;344(5). doi:10.1016/j.disc.2021.112312"},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","external_id":{"arxiv":["1808.09165"],"isi":["000633365200001"]},"article_processing_charge":"No","author":[{"last_name":"Ivanov","full_name":"Ivanov, Grigory","first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E"}],"title":"On the volume of projections of the cross-polytope","abstract":[{"text":"We study properties of the volume of projections of the n-dimensional\r\ncross-polytope $\\crosp^n = \\{ x \\in \\R^n \\mid |x_1| + \\dots + |x_n| \\leqslant 1\\}.$ We prove that the projection of $\\crosp^n$ onto a k-dimensional coordinate subspace has the maximum possible volume for k=2 and for k=3.\r\nWe obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of $\\crosp^n$ onto a k-dimensional subspace for any n>k⩾2.","lang":"eng"}],"oa_version":"Preprint","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1808.09165"}],"scopus_import":"1","intvolume":" 344","month":"05","publication_status":"published","publication_identifier":{"issn":["0012365X"]},"language":[{"iso":"eng"}],"volume":344,"issue":"5","_id":"9098","type":"journal_article","article_type":"original","status":"public","date_updated":"2023-08-07T13:40:37Z","department":[{"_id":"UlWa"}]},{"publication_identifier":{"issn":["1050-6926"],"eissn":["1559-002X"]},"publication_status":"published","language":[{"iso":"eng"}],"volume":31,"abstract":[{"text":"We extend the notion of the minimal volume ellipsoid containing a convex body in Rd to the setting of logarithmically concave functions. We consider a vast class of logarithmically concave functions whose superlevel sets are concentric ellipsoids. For a fixed function from this class, we consider the set of all its “affine” positions. For any log-concave function f on Rd, we consider functions belonging to this set of “affine” positions, and find the one with the minimal integral under the condition that it is pointwise greater than or equal to f. We study the properties of existence and uniqueness of the solution to this problem. For any s∈[0,+∞), we consider the construction dual to the recently defined John s-function (Ivanov and Naszódi in Functional John ellipsoids. arXiv preprint: arXiv:2006.09934, 2020). We prove that such a construction determines a unique function and call it the Löwner s-function of f. We study the Löwner s-functions as s tends to zero and to infinity. Finally, extending the notion of the outer volume ratio, we define the outer integral ratio of a log-concave function and give an asymptotically tight bound on it.","lang":"eng"}],"oa_version":"Preprint","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/2008.09543","open_access":"1"}],"month":"05","intvolume":" 31","date_updated":"2023-08-08T14:04:49Z","department":[{"_id":"UlWa"}],"_id":"9548","type":"journal_article","article_type":"original","status":"public","isi":1,"year":"2021","day":"31","publication":"Journal of Geometric Analysis","page":"11493-11528","date_published":"2021-05-31T00:00:00Z","doi":"10.1007/s12220-021-00691-4","date_created":"2021-06-13T22:01:32Z","acknowledgement":"The authors acknowledge the support of the grant of the Russian Government N 075-15-2019-1926.","quality_controlled":"1","publisher":"Springer","oa":1,"citation":{"chicago":"Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.” Journal of Geometric Analysis. Springer, 2021. https://doi.org/10.1007/s12220-021-00691-4.","ista":"Ivanov G, Tsiutsiurupa I. 2021. Functional Löwner ellipsoids. Journal of Geometric Analysis. 31, 11493–11528.","mla":"Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.” Journal of Geometric Analysis, vol. 31, Springer, 2021, pp. 11493–528, doi:10.1007/s12220-021-00691-4.","ama":"Ivanov G, Tsiutsiurupa I. Functional Löwner ellipsoids. Journal of Geometric Analysis. 2021;31:11493-11528. doi:10.1007/s12220-021-00691-4","apa":"Ivanov, G., & Tsiutsiurupa, I. (2021). Functional Löwner ellipsoids. Journal of Geometric Analysis. Springer. https://doi.org/10.1007/s12220-021-00691-4","ieee":"G. Ivanov and I. Tsiutsiurupa, “Functional Löwner ellipsoids,” Journal of Geometric Analysis, vol. 31. Springer, pp. 11493–11528, 2021.","short":"G. Ivanov, I. Tsiutsiurupa, Journal of Geometric Analysis 31 (2021) 11493–11528."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","author":[{"first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","last_name":"Ivanov","full_name":"Ivanov, Grigory"},{"last_name":"Tsiutsiurupa","full_name":"Tsiutsiurupa, Igor","first_name":"Igor"}],"article_processing_charge":"No","external_id":{"isi":["000656507500001"],"arxiv":["2008.09543"]},"title":"Functional Löwner ellipsoids"},{"status":"public","article_type":"original","type":"journal_article","_id":"10181","department":[{"_id":"UlWa"}],"date_updated":"2023-08-14T08:11:38Z","month":"10","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/2012.10691","open_access":"1"}],"oa_version":"Published Version","abstract":[{"text":"In this article we study some geometric properties of proximally smooth sets. First, we introduce a modification of the metric projection and prove its existence. Then we provide an algorithm for constructing a rectifiable curve between two sufficiently close points of a proximally smooth set in a uniformly convex and uniformly smooth Banach space, with the moduli of smoothness and convexity of power type. Our algorithm returns a reasonably short curve between two sufficiently close points of a proximally smooth set, is iterative and uses our modification of the metric projection. We estimate the length of the constructed curve and its deviation from the segment with the same endpoints. These estimates coincide up to a constant factor with those for the geodesics in a proximally smooth set in a Hilbert space.","lang":"eng"}],"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1877-0541"],"issn":["0927-6947"]},"publication_status":"published","title":"Rectifiable curves in proximally smooth sets","author":[{"id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","first_name":"Grigory","full_name":"Ivanov, Grigory","last_name":"Ivanov"},{"first_name":"Mariana S.","full_name":"Lopushanski, Mariana S.","last_name":"Lopushanski"}],"article_processing_charge":"No","external_id":{"arxiv":["2012.10691"],"isi":["000705774800001"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"mla":"Ivanov, Grigory, and Mariana S. Lopushanski. “Rectifiable Curves in Proximally Smooth Sets.” Set-Valued and Variational Analysis, Springer Nature, 2021, doi:10.1007/s11228-021-00612-1.","ama":"Ivanov G, Lopushanski MS. Rectifiable curves in proximally smooth sets. Set-Valued and Variational Analysis. 2021. doi:10.1007/s11228-021-00612-1","apa":"Ivanov, G., & Lopushanski, M. S. (2021). Rectifiable curves in proximally smooth sets. Set-Valued and Variational Analysis. Springer Nature. https://doi.org/10.1007/s11228-021-00612-1","ieee":"G. Ivanov and M. S. Lopushanski, “Rectifiable curves in proximally smooth sets,” Set-Valued and Variational Analysis. Springer Nature, 2021.","short":"G. Ivanov, M.S. Lopushanski, Set-Valued and Variational Analysis (2021).","chicago":"Ivanov, Grigory, and Mariana S. Lopushanski. “Rectifiable Curves in Proximally Smooth Sets.” Set-Valued and Variational Analysis. Springer Nature, 2021. https://doi.org/10.1007/s11228-021-00612-1.","ista":"Ivanov G, Lopushanski MS. 2021. Rectifiable curves in proximally smooth sets. Set-Valued and Variational Analysis."},"quality_controlled":"1","publisher":"Springer Nature","oa":1,"acknowledgement":"Theorem 2 was obtained at Steklov Mathematical Institute RAS and supported by Russian Science Foundation, grant N 19-11-00087.","date_published":"2021-10-09T00:00:00Z","doi":"10.1007/s11228-021-00612-1","date_created":"2021-10-24T22:01:35Z","day":"09","publication":"Set-Valued and Variational Analysis","isi":1,"year":"2021"},{"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ista":"Ivanov G, Tsiutsiurupa I. 2021. On the volume of sections of the cube. Analysis and Geometry in Metric Spaces. 9(1), 1–18.","chicago":"Ivanov, Grigory, and Igor Tsiutsiurupa. “On the Volume of Sections of the Cube.” Analysis and Geometry in Metric Spaces. De Gruyter, 2021. https://doi.org/10.1515/agms-2020-0103.","ama":"Ivanov G, Tsiutsiurupa I. On the volume of sections of the cube. Analysis and Geometry in Metric Spaces. 2021;9(1):1-18. doi:10.1515/agms-2020-0103","apa":"Ivanov, G., & Tsiutsiurupa, I. (2021). On the volume of sections of the cube. Analysis and Geometry in Metric Spaces. De Gruyter. https://doi.org/10.1515/agms-2020-0103","short":"G. Ivanov, I. Tsiutsiurupa, Analysis and Geometry in Metric Spaces 9 (2021) 1–18.","ieee":"G. Ivanov and I. Tsiutsiurupa, “On the volume of sections of the cube,” Analysis and Geometry in Metric Spaces, vol. 9, no. 1. De Gruyter, pp. 1–18, 2021.","mla":"Ivanov, Grigory, and Igor Tsiutsiurupa. “On the Volume of Sections of the Cube.” Analysis and Geometry in Metric Spaces, vol. 9, no. 1, De Gruyter, 2021, pp. 1–18, doi:10.1515/agms-2020-0103."},"title":"On the volume of sections of the cube","author":[{"first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","full_name":"Ivanov, Grigory","last_name":"Ivanov"},{"first_name":"Igor","last_name":"Tsiutsiurupa","full_name":"Tsiutsiurupa, Igor"}],"article_processing_charge":"No","external_id":{"isi":["000734286800001"],"arxiv":["2004.02674"]},"day":"29","publication":"Analysis and Geometry in Metric Spaces","isi":1,"has_accepted_license":"1","year":"2021","doi":"10.1515/agms-2020-0103","date_published":"2021-01-29T00:00:00Z","date_created":"2022-03-18T09:25:14Z","page":"1-18","acknowledgement":"The authors acknowledge the support of the grant of the Russian Government N 075-15-\r\n2019-1926. G.I.was supported also by the SwissNational Science Foundation grant 200021-179133. The authors are very grateful to the anonymous reviewer for valuable remarks.","publisher":"De Gruyter","quality_controlled":"1","oa":1,"ddc":["510"],"date_updated":"2023-08-17T07:07:58Z","department":[{"_id":"UlWa"}],"file_date_updated":"2022-03-18T09:31:59Z","_id":"10856","status":"public","keyword":["Applied Mathematics","Geometry and Topology","Analysis"],"article_type":"original","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"file":[{"date_updated":"2022-03-18T09:31:59Z","file_size":789801,"creator":"dernst","date_created":"2022-03-18T09:31:59Z","file_name":"2021_AnalysisMetricSpaces_Ivanov.pdf","content_type":"application/pdf","access_level":"open_access","relation":"main_file","file_id":"10857","checksum":"7e615ac8489f5eae580b6517debfdc53","success":1}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2299-3274"]},"publication_status":"published","volume":9,"issue":"1","oa_version":"Published Version","abstract":[{"lang":"eng","text":"We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1]n. We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of Rn onto a k-dimensional subspace that maximizes the volume of the intersection. We \u001cnd the optimal upper bound on the volume of a planar section of the cube [−1, 1]n , n ≥ 2."}],"month":"01","intvolume":" 9","scopus_import":"1"},{"status":"public","keyword":["General Mathematics","Tight frame","Grassmannian","zonotope"],"type":"journal_article","article_type":"original","_id":"10860","department":[{"_id":"UlWa"}],"date_updated":"2023-09-05T12:43:09Z","month":"12","intvolume":" 64","scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1804.10055"}],"oa_version":"Preprint","abstract":[{"lang":"eng","text":"A tight frame is the orthogonal projection of some orthonormal basis of Rn onto Rk. We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary condition for local extrema of these problems. As applications, we prove new results for the problem of maximization of the volume of zonotopes."}],"issue":"4","volume":64,"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1496-4287"],"issn":["0008-4395"]},"publication_status":"published","title":"Tight frames and related geometric problems","author":[{"full_name":"Ivanov, Grigory","last_name":"Ivanov","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E","first_name":"Grigory"}],"external_id":{"isi":["000730165300021"],"arxiv":["1804.10055"]},"article_processing_charge":"No","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"ama":"Ivanov G. Tight frames and related geometric problems. Canadian Mathematical Bulletin. 2021;64(4):942-963. doi:10.4153/s000843952000096x","apa":"Ivanov, G. (2021). Tight frames and related geometric problems. Canadian Mathematical Bulletin. Canadian Mathematical Society. https://doi.org/10.4153/s000843952000096x","ieee":"G. Ivanov, “Tight frames and related geometric problems,” Canadian Mathematical Bulletin, vol. 64, no. 4. Canadian Mathematical Society, pp. 942–963, 2021.","short":"G. Ivanov, Canadian Mathematical Bulletin 64 (2021) 942–963.","mla":"Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” Canadian Mathematical Bulletin, vol. 64, no. 4, Canadian Mathematical Society, 2021, pp. 942–63, doi:10.4153/s000843952000096x.","ista":"Ivanov G. 2021. Tight frames and related geometric problems. Canadian Mathematical Bulletin. 64(4), 942–963.","chicago":"Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” Canadian Mathematical Bulletin. Canadian Mathematical Society, 2021. https://doi.org/10.4153/s000843952000096x."},"publisher":"Canadian Mathematical Society","quality_controlled":"1","oa":1,"acknowledgement":"The author was supported by the Swiss National Science Foundation grant 200021_179133. The author acknowledges the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no. 075-15-2019-1926.","date_published":"2021-12-18T00:00:00Z","doi":"10.4153/s000843952000096x","date_created":"2022-03-18T09:55:59Z","page":"942-963","day":"18","publication":"Canadian Mathematical Bulletin","isi":1,"year":"2021"}]