{"year":"2023","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."}],"oa":1,"page":"20613-20669","author":[{"first_name":"Grigory","last_name":"Ivanov","full_name":"Ivanov, Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E"},{"full_name":"Naszódi, Márton","last_name":"Naszódi","first_name":"Márton"}],"day":"01","language":[{"iso":"eng"}],"tmp":{"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)","image":"/images/cc_by_nc_nd.png"},"has_accepted_license":"1","publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"ddc":["510"],"title":"Functional John and Löwner conditions for pairs of log-concave functions","citation":{"short":"G. Ivanov, M. Naszódi, International Mathematics Research Notices 2023 (2023) 20613–20669.","apa":"Ivanov, G., &#38; Naszódi, M. (2023). Functional John and Löwner conditions for pairs of log-concave functions. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnad210\">https://doi.org/10.1093/imrn/rnad210</a>","chicago":"Ivanov, Grigory, and Márton Naszódi. “Functional John and Löwner Conditions for Pairs of Log-Concave Functions.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2023. <a href=\"https://doi.org/10.1093/imrn/rnad210\">https://doi.org/10.1093/imrn/rnad210</a>.","ama":"Ivanov G, Naszódi M. Functional John and Löwner conditions for pairs of log-concave functions. <i>International Mathematics Research Notices</i>. 2023;2023(23):20613-20669. doi:<a href=\"https://doi.org/10.1093/imrn/rnad210\">10.1093/imrn/rnad210</a>","ieee":"G. Ivanov and M. Naszódi, “Functional John and Löwner conditions for pairs of log-concave functions,” <i>International Mathematics Research Notices</i>, 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.” <i>International Mathematics Research Notices</i>, vol. 2023, no. 23, Oxford University Press, 2023, pp. 20613–69, doi:<a href=\"https://doi.org/10.1093/imrn/rnad210\">10.1093/imrn/rnad210</a>."},"intvolume":"      2023","article_processing_charge":"Yes (via OA deal)","date_published":"2023-12-01T00:00:00Z","volume":2023,"quality_controlled":"1","external_id":{"arxiv":["2212.11781"]},"publisher":"Oxford University Press","publication_status":"published","month":"12","date_created":"2024-01-08T09:48:56Z","issue":"23","publication":"International Mathematics Research Notices","date_updated":"2024-01-08T09:57:25Z","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.1093/imrn/rnad210","oa_version":"Published Version","department":[{"_id":"UlWa"}],"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.].","file_date_updated":"2024-01-08T09:53:09Z","file":[{"creator":"dernst","access_level":"open_access","success":1,"date_updated":"2024-01-08T09:53:09Z","file_size":815777,"date_created":"2024-01-08T09:53:09Z","checksum":"353666cea80633beb0f1ffd342dff6d4","file_name":"2023_IMRN_Ivanov.pdf","content_type":"application/pdf","relation":"main_file","file_id":"14738"}],"_id":"14737","article_type":"original","status":"public","keyword":["General Mathematics"]}