论文标题
功能性约翰·埃利普斯（John Ellipsoids）
Functional John Ellipsoids
论文作者
论文摘要
我们引入了一种在$ \ mathbb {r}^{d} $上表示对数凹入函数的新方法。它使我们能够将凸体中包含的最大体积椭球的概念扩展到对数凹入函数的设置，如下所示。对于每一个$ s> 0 $，我们在$ \ mathbb {r}^{d} $上定义一类非负函数，从$ \ mathbb {r}^{d+1} $中衍生出的。对于$ \ mathbb {r}^{d} $上的任何log-concave函数$ f $，以及任何固定的$ s> 0 $，我们考虑属于该类别的函数，并找到具有最大积分的功能，条件下它却小于或等于$ f $，并且我们称其为\ emph {\ emph {\ jsfunuttion。在建立存在和独特性之后，我们给出了这种功能的特征，类似于约翰在其基本定理中给出的功能。我们发现John $ s $ functions会趋于椭圆形的特征功能，因为$ s $倾向于零，而高斯密度为$ s $倾向于无穷大。 作为一种应用程序，我们证明了一个定量的Helly类型结果：任何log-concove函数的最低点的积分至少是一个常数$ C_D $多数的集成数的不可或缺的最小值的最小值的$ 3D+2 $的最低限度的最低限度，而$ c_d $仅取决于$ d $ $ d $。
We introduce a new way of representing logarithmically concave functions on $\mathbb{R}^{d}$. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every $s>0$, we define a class of non-negative functions on $\mathbb{R}^{d}$ derived from ellipsoids in $\mathbb{R}^{d+1}$. For any log-concave function $f$ on $\mathbb{R}^{d}$, and any fixed $s>0$, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to $f$, and we call it the \emph{\jsfunction} of $f$. After establishing existence and uniqueness, we give a characterization of this function similar to the one given by John in his fundamental theorem. We find that John $s$-functions converge to characteristic functions of ellipsoids as $s$ tends to zero and to Gaussian densities as $s$ tends to infinity. As an application, we prove a quantitative Helly type result: the integral of the pointwise minimum of any family of log-concave functions is at least a constant $c_d$ multiple of the integral of the pointwise minimum of a properly chosen subfamily of size $3d+2$, where $c_d$ depends only on $d$.
