We extend the notion of the minimal volume ellipsoid containing a convex body in \(\mathbb {R}^{d}\) 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 \(\mathbb {R}^{d},\) 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 \(s \in [0,+\infty ),\) 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.

我们将包含\(\mathbb {R}^{d}\)中的凸体的最小体积椭球的概念扩展 到对数凹函数的设置。我们考虑一大类对数凹函数，其超级集是同心椭球。对于此类的固定函数，我们考虑其所有“仿射”位置的集合。对于\(\mathbb {R}^{d},\)上的任何对数凹函数f ，我们考虑属于这组“仿射”位置的函数，并在它是逐点的条件下找到具有最小积分的函数大于或等于f。我们研究这个问题解的存在性和唯一性的性质。对于任何\(s \in [0,+\infty ),\)我们考虑对最近定义的 John s函数的构造对偶（Ivanov 和 Naszódi in Functional John ellipsoids. arXiv preprint: arXiv:2006.09934, 2020）。我们证明了这种结构决定了一个独特的功能，并把它称为的L owner S函数的˚F。我们研究 Löwner s函数，因为s趋于零和无穷大。最后，扩展外体积比的概念，我们定义对数凹函数的外积分比，并在其上给出渐近紧界。