论文标题
功能性的John和Löwner条件，用于对数凸函数对
Functional John and Löwner conditions for pairs of log-concave functions
论文作者
论文摘要
John的基本定理是凸面主体中包含的最大体积椭圆形的$ k $ in $ \ mathbb {r}^d $中的最大椭圆形的定理，已经看到了几种概括和扩展。 V. Milman发起的一个方向是用另一个身体$ L $的位置（仿射图像）代替椭圆形。另一个最新的方向是在$ \ mathbb {r}^d $而不是凸面上考虑对数凹面的功能：我们将一些特殊的，径向的对称的对数concove $ g $ g $作为欧几里得球的类似物，并希望在某些给定的logcove $ $ $ $ concove $ $ $上找到最大的积分位置。 我们同时遵循这两个方向：我们考虑功能问题，并允许任何有意义的功能在上面扮演$ g $的角色。我们的一般定理在两个方向上共同扩展已知结果。 凸面设置的双重问题要求最小的椭圆形，称为\ emph {l {Ö} wner的椭圆形}，包含$ k $。我们考虑功能的类似问题：我们表征了优化问题的解决方案，即在$ f $以上的限制下，找到某些日志concove函数$ g $的最小积分位置$ g $。事实证明，在功能环境中，约翰和l {Ö}问题之间的关系比在凸体的环境中更复杂。
John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^d$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^d$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called \emph{L{ö}wner's ellipsoid}, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the L{ö}wner problems is more intricate than it is in the setting of convex bodies.
