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In this article we show the following result: if $C$ is an $n$-dimensionalconvex 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 thesupremum \[ \sup_{C,f}\int_C\phi(f(x))dx, \] where the supremum ranges over allsets $C$ 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 somein [Thm. 1.2, GoMe]. Besides, we also obtain some new estimates on the volumeof particular sections of a convex set $K$ passing through a new point of $K$.

Estimating the average of functions with convexity properties by means of a new center