论文标题
功能性Löwner椭圆形
Functional Löwner Ellipsoids
论文作者
论文摘要
我们将包含凸体的最小体积椭圆形的概念扩展到〜$ \ mathbb {r}^{d} $中的凸面，并将其设置为对数的凹入函数。我们考虑了一类巨大的对数凹面函数，其超级套件是同心椭圆形。对于此类的固定功能，我们考虑其所有“仿射”位置的集合。对于$ \ Mathbb {r}^{d}上的任何log-concove函数$ f $，我们考虑属于此“仿射”位置的函数，并找到具有最小积分的条件下的函数，条件下它大于或等于$ f。对于[0，\ infty）中的任何$ s \，我们考虑了最近定义的John $ s $ function \ cite {ivanov2020Functunctions}的构造双重。我们证明，这样的构造决定了一个独特的功能，并将其称为$ f。$的\ emph {löwner$ s $ function}，我们研究löwner$ s $ functions $ s $倾向于零且无限。最后，扩展了外部体积比的概念，我们定义了对数符号函数的外部积分比，并在其上给出了渐近的紧密结合。 \ end {摘要}
We extend the notion of the smallest volume ellipsoid containing a convex body in~$\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}
