Ivanov, GrigoryISTA; Naszódi, Márton
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. As 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.
Journal of Functional Analysis
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.
Ivanov G, Naszódi M. Functional John ellipsoids. Journal of Functional Analysis. 2022;282(11). doi:10.1016/j.jfa.2022.109441
Ivanov, G., & Naszódi, M. (2022). Functional John ellipsoids. Journal of Functional Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2022.109441
Ivanov, Grigory, and Márton Naszódi. “Functional John Ellipsoids.” Journal of Functional Analysis. Elsevier, 2022. https://doi.org/10.1016/j.jfa.2022.109441.
G. Ivanov and M. Naszódi, “Functional John ellipsoids,” Journal of Functional Analysis, vol. 282, no. 11. Elsevier, 2022.
Ivanov G, Naszódi M. 2022. Functional John ellipsoids. Journal of Functional Analysis. 282(11), 109441.
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.