  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 \[ \sup_{C,f}\int_C\phi(f(x))dx, \] 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 $K$.
