--- _id: '12680' abstract: - lang: eng 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. article_number: '113363' article_processing_charge: No article_type: letter_note author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Seyda full_name: Köse, Seyda id: 8ba3170d-dc85-11ea-9058-c4251c96a6eb last_name: Köse citation: 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 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 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. ieee: G. Ivanov and S. Köse, “Erdős-Ko-Rado and Hilton-Milner theorems for two-forms,” Discrete Mathematics, vol. 346, no. 6. Elsevier, 2023. ista: Ivanov G, Köse S. 2023. Erdős-Ko-Rado and Hilton-Milner theorems for two-forms. Discrete Mathematics. 346(6), 113363. 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. short: G. Ivanov, S. Köse, Discrete Mathematics 346 (2023). date_created: 2023-02-26T23:01:00Z date_published: 2023-06-01T00:00:00Z date_updated: 2023-10-04T11:54:57Z day: '01' department: - _id: UlWa - _id: GradSch doi: 10.1016/j.disc.2023.113363 external_id: arxiv: - '2201.10892' intvolume: ' 346' issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: ' https://doi.org/10.48550/arXiv.2201.10892' month: '06' oa: 1 oa_version: Preprint publication: Discrete Mathematics publication_identifier: issn: - 0012-365X publication_status: published publisher: Elsevier quality_controlled: '1' related_material: record: - id: '13331' relation: dissertation_contains status: public scopus_import: '1' status: public title: Erdős-Ko-Rado and Hilton-Milner theorems for two-forms type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 346 year: '2023' ... --- _id: '14660' abstract: - lang: eng text: "The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set \U0001D446⊂ℝ\U0001D451, then there are at most 2\U0001D451 points of \U0001D446 whose convex hull contains the origin in the interior. Bárány, Katchalski,and Pach proved the following quantitative version of Steinitz’s theorem. Let \U0001D444 be a convex polytope in ℝ\U0001D451 containing the standard Euclidean unit ball \U0001D401\U0001D451. Then there exist at most 2\U0001D451 vertices of \U0001D444 whose convex hull \U0001D444′ satisfies \U0001D45F\U0001D401\U0001D451⊂\U0001D444′ with \U0001D45F⩾\U0001D451−2\U0001D451. They conjectured that \U0001D45F⩾\U0001D450\U0001D451−1∕2 holds with a universal constant \U0001D450>0. We prove \U0001D45F⩾15\U0001D4512, the first polynomial lower bound on \U0001D45F. Furthermore, we show that \U0001D45F is not greater than 2/√\U0001D451." 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) article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Márton full_name: Naszódi, Márton last_name: Naszódi 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' 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.' ieee: 'G. Ivanov and M. Naszódi, “Quantitative Steinitz theorem: A polynomial bound,” Bulletin of the London Mathematical Society. London Mathematical Society, 2023.' ista: 'Ivanov G, Naszódi M. 2023. Quantitative Steinitz theorem: A polynomial bound. Bulletin of the London Mathematical Society.' 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.' short: G. Ivanov, M. Naszódi, Bulletin of the London Mathematical Society (2023). date_created: 2023-12-10T23:00:58Z date_published: 2023-12-04T00:00:00Z date_updated: 2023-12-11T10:03:54Z day: '04' department: - _id: UlWa doi: 10.1112/blms.12965 external_id: arxiv: - '2212.04308' language: - iso: eng main_file_link: - open_access: '1' url: ' https://doi.org/10.1112/blms.12965' month: '12' oa: 1 oa_version: Published Version publication: Bulletin of the London Mathematical Society publication_identifier: eissn: - 1469-2120 issn: - 0024-6093 publication_status: epub_ahead publisher: London Mathematical Society quality_controlled: '1' scopus_import: '1' status: public title: 'Quantitative Steinitz theorem: A polynomial bound' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2023' ... --- _id: '14737' 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.' 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.]." article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Márton full_name: Naszódi, Márton last_name: Naszódi citation: 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 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. 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. 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. short: G. Ivanov, M. Naszódi, International Mathematics Research Notices 2023 (2023) 20613–20669. date_created: 2024-01-08T09:48:56Z date_published: 2023-12-01T00:00:00Z date_updated: 2024-01-08T09:57:25Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.1093/imrn/rnad210 external_id: arxiv: - '2212.11781' file: - access_level: open_access checksum: 353666cea80633beb0f1ffd342dff6d4 content_type: application/pdf creator: dernst date_created: 2024-01-08T09:53:09Z date_updated: 2024-01-08T09:53:09Z file_id: '14738' file_name: 2023_IMRN_Ivanov.pdf file_size: 815777 relation: main_file success: 1 file_date_updated: 2024-01-08T09:53:09Z has_accepted_license: '1' intvolume: ' 2023' issue: '23' keyword: - General Mathematics language: - iso: eng license: https://creativecommons.org/licenses/by-nc-nd/4.0/ month: '12' oa: 1 oa_version: Published Version page: 20613-20669 publication: International Mathematics Research Notices publication_identifier: eissn: - 1687-0247 issn: - 1073-7928 publication_status: published publisher: Oxford University Press quality_controlled: '1' status: public title: Functional John and Löwner conditions for pairs of log-concave functions tmp: image: /images/cc_by_nc_nd.png legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) short: CC BY-NC-ND (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 2023 year: '2023' ... --- _id: '10887' 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." 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. ' article_number: '109441' article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Márton full_name: Naszódi, Márton last_name: Naszódi citation: ama: Ivanov G, Naszódi M. Functional John ellipsoids. Journal of Functional Analysis. 2022;282(11). doi:10.1016/j.jfa.2022.109441 apa: Ivanov, G., & Naszódi, M. (2022). Functional John ellipsoids. Journal of Functional Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2022.109441 chicago: Ivanov, Grigory, and Márton Naszódi. “Functional John Ellipsoids.” Journal of Functional Analysis. Elsevier, 2022. https://doi.org/10.1016/j.jfa.2022.109441. ieee: G. Ivanov and M. Naszódi, “Functional John ellipsoids,” Journal of Functional Analysis, vol. 282, no. 11. Elsevier, 2022. ista: Ivanov G, Naszódi M. 2022. Functional John ellipsoids. Journal of Functional Analysis. 282(11), 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. short: G. Ivanov, M. Naszódi, Journal of Functional Analysis 282 (2022). date_created: 2022-03-20T23:01:38Z date_published: 2022-06-01T00:00:00Z date_updated: 2023-08-02T14:51:11Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.1016/j.jfa.2022.109441 external_id: arxiv: - '2006.09934' isi: - '000781371300008' file: - access_level: open_access checksum: 1cf185e264e04c87cb1ce67a00db88ab content_type: application/pdf creator: dernst date_created: 2022-08-02T10:40:48Z date_updated: 2022-08-02T10:40:48Z file_id: '11721' file_name: 2022_JourFunctionalAnalysis_Ivanov.pdf file_size: 734482 relation: main_file success: 1 file_date_updated: 2022-08-02T10:40:48Z has_accepted_license: '1' intvolume: ' 282' isi: 1 issue: '11' language: - iso: eng month: '06' oa: 1 oa_version: Published Version publication: Journal of Functional Analysis publication_identifier: eissn: - 1096-0783 issn: - 0022-1236 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Functional John ellipsoids tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 282 year: '2022' ... --- _id: '11435' abstract: - lang: eng 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 $Q \subset - 8d^3 Q^{\prime \prime}.$' 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." article_processing_charge: No article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Marton full_name: Naszodi, Marton last_name: Naszodi citation: 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' 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' 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.' ista: 'Ivanov G, Naszodi M. 2022. A quantitative Helly-type theorem: Containment in a homothet. SIAM Journal on Discrete Mathematics. 36(2), 951–957.' 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.' short: G. Ivanov, M. Naszodi, SIAM Journal on Discrete Mathematics 36 (2022) 951–957. date_created: 2022-06-05T22:01:50Z date_published: 2022-04-11T00:00:00Z date_updated: 2023-10-18T06:58:03Z day: '11' department: - _id: UlWa doi: 10.1137/21M1403308 external_id: arxiv: - '2103.04122' isi: - '000793158200002' intvolume: ' 36' isi: 1 issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: ' https://doi.org/10.48550/arXiv.2103.04122' month: '04' oa: 1 oa_version: Preprint page: 951-957 publication: SIAM Journal on Discrete Mathematics publication_identifier: issn: - 0895-4801 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: 'A quantitative Helly-type theorem: Containment in a homothet' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 36 year: '2022' ... --- _id: '9037' abstract: - lang: eng 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 \U0001D700 ‐net theorem in Banach spaces of type \U0001D45D>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." 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." article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov citation: 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 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 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. 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. 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. date_created: 2021-01-24T23:01:08Z date_published: 2021-04-01T00:00:00Z date_updated: 2023-08-07T13:35:20Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.1112/blms.12449 external_id: arxiv: - '1912.08561' isi: - '000607265100001' file: - access_level: open_access checksum: e6ceaa6470d835eb4c211cbdd38fdfd1 content_type: application/pdf creator: kschuh date_created: 2021-08-06T09:59:45Z date_updated: 2021-08-06T09:59:45Z file_id: '9796' file_name: 2021_BLMS_Ivanov.pdf file_size: 194550 relation: main_file success: 1 file_date_updated: 2021-08-06T09:59:45Z has_accepted_license: '1' intvolume: ' 53' isi: 1 issue: '2' language: - iso: eng month: '04' oa: 1 oa_version: Published Version page: 631-641 publication: Bulletin of the London Mathematical Society publication_identifier: eissn: - '14692120' issn: - '00246093' publication_status: published publisher: London Mathematical Society quality_controlled: '1' scopus_import: '1' status: public title: No-dimension Tverberg's theorem and its corollaries in Banach spaces of type p tmp: image: /images/cc_by_nc_nd.png legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) short: CC BY-NC-ND (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 53 year: '2021' ... --- _id: '9098' abstract: - lang: eng 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." 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). article_number: '112312' article_processing_charge: No article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov citation: ama: Ivanov G. On the volume of projections of the cross-polytope. Discrete Mathematics. 2021;344(5). doi:10.1016/j.disc.2021.112312 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 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. ieee: G. Ivanov, “On the volume of projections of the cross-polytope,” Discrete Mathematics, vol. 344, no. 5. Elsevier, 2021. 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). date_created: 2021-02-07T23:01:12Z date_published: 2021-05-01T00:00:00Z date_updated: 2023-08-07T13:40:37Z day: '01' department: - _id: UlWa doi: 10.1016/j.disc.2021.112312 external_id: arxiv: - '1808.09165' isi: - '000633365200001' intvolume: ' 344' isi: 1 issue: '5' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1808.09165 month: '05' oa: 1 oa_version: Preprint publication: Discrete Mathematics publication_identifier: issn: - 0012365X publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: On the volume of projections of the cross-polytope type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 344 year: '2021' ... --- _id: '9548' abstract: - lang: eng 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.' acknowledgement: The authors acknowledge the support of the grant of the Russian Government N 075-15-2019-1926. article_processing_charge: No article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Igor full_name: Tsiutsiurupa, Igor last_name: Tsiutsiurupa citation: 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 chicago: Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.” Journal of Geometric Analysis. Springer, 2021. 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. 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. short: G. Ivanov, I. Tsiutsiurupa, Journal of Geometric Analysis 31 (2021) 11493–11528. date_created: 2021-06-13T22:01:32Z date_published: 2021-05-31T00:00:00Z date_updated: 2023-08-08T14:04:49Z day: '31' department: - _id: UlWa doi: 10.1007/s12220-021-00691-4 external_id: arxiv: - '2008.09543' isi: - '000656507500001' intvolume: ' 31' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2008.09543 month: '05' oa: 1 oa_version: Preprint page: 11493-11528 publication: Journal of Geometric Analysis publication_identifier: eissn: - 1559-002X issn: - 1050-6926 publication_status: published publisher: Springer quality_controlled: '1' scopus_import: '1' status: public title: Functional Löwner ellipsoids type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 31 year: '2021' ... --- _id: '10181' abstract: - lang: eng 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. acknowledgement: Theorem 2 was obtained at Steklov Mathematical Institute RAS and supported by Russian Science Foundation, grant N 19-11-00087. article_processing_charge: No article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Mariana S. full_name: Lopushanski, Mariana S. last_name: Lopushanski citation: 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 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. ieee: G. Ivanov and M. S. Lopushanski, “Rectifiable curves in proximally smooth sets,” Set-Valued and Variational Analysis. Springer Nature, 2021. ista: Ivanov G, Lopushanski MS. 2021. Rectifiable curves in proximally smooth sets. Set-Valued and Variational Analysis. 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. short: G. Ivanov, M.S. Lopushanski, Set-Valued and Variational Analysis (2021). date_created: 2021-10-24T22:01:35Z date_published: 2021-10-09T00:00:00Z date_updated: 2023-08-14T08:11:38Z day: '09' department: - _id: UlWa doi: 10.1007/s11228-021-00612-1 external_id: arxiv: - '2012.10691' isi: - '000705774800001' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2012.10691 month: '10' oa: 1 oa_version: Published Version publication: Set-Valued and Variational Analysis publication_identifier: eissn: - 1877-0541 issn: - 0927-6947 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Rectifiable curves in proximally smooth sets type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 year: '2021' ... --- _id: '10856' 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 \x1Cnd the optimal upper bound on the volume of a planar section of the cube [−1, 1]n , n ≥ 2." 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." article_processing_charge: No article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov - first_name: Igor full_name: Tsiutsiurupa, Igor last_name: Tsiutsiurupa citation: 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 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. 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. ista: Ivanov G, Tsiutsiurupa I. 2021. On the volume of sections of the cube. Analysis and Geometry in Metric Spaces. 9(1), 1–18. 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. short: G. Ivanov, I. Tsiutsiurupa, Analysis and Geometry in Metric Spaces 9 (2021) 1–18. date_created: 2022-03-18T09:25:14Z date_published: 2021-01-29T00:00:00Z date_updated: 2023-08-17T07:07:58Z day: '29' ddc: - '510' department: - _id: UlWa doi: 10.1515/agms-2020-0103 external_id: arxiv: - '2004.02674' isi: - '000734286800001' file: - access_level: open_access checksum: 7e615ac8489f5eae580b6517debfdc53 content_type: application/pdf creator: dernst date_created: 2022-03-18T09:31:59Z date_updated: 2022-03-18T09:31:59Z file_id: '10857' file_name: 2021_AnalysisMetricSpaces_Ivanov.pdf file_size: 789801 relation: main_file success: 1 file_date_updated: 2022-03-18T09:31:59Z has_accepted_license: '1' intvolume: ' 9' isi: 1 issue: '1' keyword: - Applied Mathematics - Geometry and Topology - Analysis language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 1-18 publication: Analysis and Geometry in Metric Spaces publication_identifier: issn: - 2299-3274 publication_status: published publisher: De Gruyter quality_controlled: '1' scopus_import: '1' status: public title: On the volume of sections of the cube tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 9 year: '2021' ... --- _id: '10860' 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. 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. article_processing_charge: No article_type: original author: - first_name: Grigory full_name: Ivanov, Grigory id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E last_name: Ivanov 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 chicago: Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” Canadian Mathematical Bulletin. Canadian Mathematical Society, 2021. 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. ista: Ivanov G. 2021. Tight frames and related geometric problems. Canadian Mathematical Bulletin. 64(4), 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. short: G. Ivanov, Canadian Mathematical Bulletin 64 (2021) 942–963. date_created: 2022-03-18T09:55:59Z date_published: 2021-12-18T00:00:00Z date_updated: 2023-09-05T12:43:09Z day: '18' department: - _id: UlWa doi: 10.4153/s000843952000096x external_id: arxiv: - '1804.10055' isi: - '000730165300021' intvolume: ' 64' isi: 1 issue: '4' keyword: - General Mathematics - Tight frame - Grassmannian - zonotope language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1804.10055 month: '12' oa: 1 oa_version: Preprint page: 942-963 publication: Canadian Mathematical Bulletin publication_identifier: eissn: - 1496-4287 issn: - 0008-4395 publication_status: published publisher: Canadian Mathematical Society quality_controlled: '1' scopus_import: '1' status: public title: Tight frames and related geometric problems type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 64 year: '2021' ...