Ivanov, GrigoryISTA; Tsiutsiurupa, Igor
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.
Journal of Geometric Analysis
The authors acknowledge the support of the grant of the Russian Government N 075-15-2019-1926.
Ivanov G, Tsiutsiurupa I. Functional Löwner ellipsoids. Journal of Geometric Analysis. 2021;31:11493-11528. doi:10.1007/s12220-021-00691-4
Ivanov, G., & Tsiutsiurupa, I. (2021). Functional Löwner ellipsoids. Journal of Geometric Analysis. Springer. https://doi.org/10.1007/s12220-021-00691-4
Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.” Journal of Geometric Analysis. Springer, 2021. https://doi.org/10.1007/s12220-021-00691-4.
G. Ivanov and I. Tsiutsiurupa, “Functional Löwner ellipsoids,” Journal of Geometric Analysis, vol. 31. Springer, pp. 11493–11528, 2021.
Ivanov G, Tsiutsiurupa I. 2021. Functional Löwner ellipsoids. Journal of Geometric Analysis. 31, 11493–11528.
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.