论文标题
通过新中心估算具有凸性属性的函数的平均值
Estimating the average of functions with convexity properties by means of a new center
论文作者
论文摘要
In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $ϕ:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $ϕ(0)=0$, we then characterize the class of sets and concave functions that attain the supemum \ [\ sup_ {c，f} \ int_cx（f（x））dx，\]其中，上级范围范围范围在所有集合上，$ n $ dimementional量$ | c | c | c | = c | = c $ and $ f（x_ {c，f}）= k $ in Point $ x____________的其他条件。两个非负常量$ c，k> 0 $。 结果，我们在[MP]中扩展了Milman和Pajor的一些结果，在[THM]中扩展了一些结果。 1.2，戈姆]。此外，我们还获得了一些新的估算值，以了解凸套$ k $的特定部分的量。
In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $ϕ:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $ϕ(0)=0$, we then characterize the class of sets and concave functions that attain the supremum \[ \sup_{C,f}\int_Cϕ(f(x))dx, \] where the supremum ranges over all sets $C$ with $n$-dimensional volume $|C|=c$ and the additional condition that $f(x_{C,f})=k$ for some point $x_{C,f}\in C$ that we introduce in the article, for two non-negative constants $c,k>0$. As a consequence, we extend some results of Milman and Pajor in [MP] and some in [Thm. 1.2, GoMe]. Besides, we also obtain some new estimates on the volume of particular sections of a convex set $K$ passing through a new point of $K$.
