泛函Löwner椭球
Functional Löwner Ellipsoids
摘要：
本文将~$\mathbb{R}^{d}$中包含凸体的最小体积椭球的概念推广到数学上凹函数的设置。我们考虑了一类极大的数学上的凹函数，其超水平集是同心椭球。对于一个固定的功能从这个类，我们认为所有的“仿射”的位置。对于$\mathbb{R}^{d},$上的对数凹函数$f$，我们考虑属于这组"仿射"位置的函数，并在逐点大于或等于$f.$的条件下，找到积分最小的函数。对于任何$s \in [0,\infty),$，我们认为该构造与最近定义的John $s$-函数\cite{ivanov2020functional}是对偶的。我们证明，这样的建设确定一个唯一的功能，并称之为\emph{Löwner $s$-function}的$f.$我们研究的Löwner $s$-功能，$s$趋于零和无穷大。最后，推广了外体积比的概念，定义了对数凹函数的外积分比，并给出了它的渐近紧界. \end{abstract}
英文摘要：
We extend the notion of the smallest 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 smallest 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 \cite{ivanov2020functional}. We prove that such a construction determines a unique function and call it the \emph{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. \end{abstract}
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