article published yes Grigory Ivanov author 87744F66-5C6F-11EA-AFE0-D16B3DDC885E Igor Tsiutsiurupa author UlWa department 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. Springer2021 eng 1050-6926 1559-002X 2008.09543 00065650750000110.1007/s12220-021-00691-4 3111493-11528 G. Ivanov and I. Tsiutsiurupa, “Functional Löwner ellipsoids,” <i>Journal of Geometric Analysis</i>, vol. 31. Springer, pp. 11493–11528, 2021. Ivanov G, Tsiutsiurupa I. Functional Löwner ellipsoids. <i>Journal of Geometric Analysis</i>. 2021;31:11493-11528. doi:<a href="https://doi.org/10.1007/s12220-021-00691-4">10.1007/s12220-021-00691-4</a> G. Ivanov, I. Tsiutsiurupa, Journal of Geometric Analysis 31 (2021) 11493–11528. Ivanov, G., &#38; Tsiutsiurupa, I. (2021). Functional Löwner ellipsoids. <i>Journal of Geometric Analysis</i>. Springer. <a href="https://doi.org/10.1007/s12220-021-00691-4">https://doi.org/10.1007/s12220-021-00691-4</a> Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.” <i>Journal of Geometric Analysis</i>, vol. 31, Springer, 2021, pp. 11493–528, doi:<a href="https://doi.org/10.1007/s12220-021-00691-4">10.1007/s12220-021-00691-4</a>. Ivanov G, Tsiutsiurupa I. 2021. Functional Löwner ellipsoids. Journal of Geometric Analysis. 31, 11493–11528. Ivanov, Grigory, and Igor Tsiutsiurupa. “Functional Löwner Ellipsoids.” <i>Journal of Geometric Analysis</i>. Springer, 2021. <a href="https://doi.org/10.1007/s12220-021-00691-4">https://doi.org/10.1007/s12220-021-00691-4</a>. 95482021-06-13T22:01:32Z2023-08-08T14:04:49Z