泛函约翰椭球
Functional John Ellipsoids
摘要：
介绍了一种在$\mathbb{R}^{d}$上表示数学凹函数的新方法。它使我们能够扩展的概念，最大的体积椭球包含在凸体的设置几何凹函数如下。对于每个$s>0$，我们定义了一类由$\mathbb{R}^{d+1}$中的椭球导出的$\mathbb{R}^{d}$上的非负函数。对$\mathbb{R}^{d}$上的对数凹函数$f$，以及固定的$s>0$，我们考虑属于这类的函数，在逐点小于等于$f$的条件下，找到积分最大的一个，我们称它为$f$的\emph{\jsfunction}.在建立存在性和唯一性之后，我们给出了这个函数的一个类似于约翰在他的基本定理中给出的一个的特征。当$s$趋于零时，John $s$ -函数收敛于椭球的特征函数;当$s$趋于无穷大时，John函数收敛于高斯密度.作为应用，我们证明了一个定量的Helly型结果：任何对数凹函数族的点态极小值的积分至少是适当选择的大小为$3d+2$的子族的点态极小值的积分的常数$c_d$倍，其中$c_d$仅依赖于$d$ .
英文摘要：
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 $d$.
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