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

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 $K$.