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 \(\phi :[0,\infty )\rightarrow [0,\infty )\) is a convex function with \(\phi (0)=0\), we then characterize the class of sets and concave functions that attain the supremum

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 nonnegative constants \(c,k>0\). As a consequence, we extend some results of Milman and Pajor in (Adv Math 152(2):314–335, 2000) and some in (Mediterr J Math, 2020, Thm. 1.2). 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.

在本文中，我们显示以下结果：如果C是n维凸和紧子集，\（f：C \ rightarrow [0，\ infty} \）是凹面，而\（\ phi：[0，\ infty）\ rightarrow [0，\ infty）\）是具有\（\ phi（0）= 0 \）的凸函数，然后我们描述达到最高的集合和凹函数的类别

其中在所有集合C上具有n维空间\（| C | = c \）的最高值，并且对于某个点\（x_ {C ，f} \在本文中介绍的C \）中，用于两个非负常量\（c，k> 0 \）。结果，我们将Milman和Pajor的一些结果扩展到（Adv Math 152（2）：314-335，2000）中，并扩展一些结果（Mediterr J Math，2020，Thm。1.2）。此外，我们还获得了关于凸集K穿过新点K的特定部分的体积的一些新估计。