Jensen-type geometric shapes

We present both necessary and sufficient conditions to the convex closed shape $X$ such that the inequality $$\frac{1}{\mathrm{\mid }X\mathrm{\mid }}{\int }_{X}f(x)dx\le \frac{1}{\mathrm{\mid }\mathrm{\partial }X\mathrm{\mid }}{\int }_{\mathrm{\partial }X}f(x)dx$$ is valid for every convex function $f\text{\hspace{0.17em}⁣}:X\to \mathbb{R}$ ($\mathrm{\partial }X$ stands for the boundary of $X$). It is proved that this inequality holds if $X$ is (i) an $n$-dimensional parallelotope, (ii) an $n$-dimensional ball, (iii) a convex polytope having an inscribed sphere (tangent to all its facets) with center in the center of mass of $\mathrm{\partial }X$.