The Hermite-Hadamard inequality on hypercuboid

Given any $\mathbf{a}:=\left(a_1,a_2,\dots ,a_n\right)$ and $\mathbf{b}:=\left(b_1,b_2,\dots ,b_n\right)$ in ${\mathbb{R}}^n$. The $\mathbf{\text{n}}$-fold convex function defined on $\left[ {{\bf{a}},{\bf{b}}} \right]$,${\bf{a}},{\bf{b}} \in $\mathbb{R}$^n$with${\bf{a}}<{\bf{b}}$ is a convex function in each variable separately. In this work we prove an inequality of Hermite-Hadamard type for $\mathbf{\text{n}}$-fold convex functions. Namely, we establish the inequality \begin{align*} f\left( {\frac{{{\bf{a}} + {\bf{b}}}}{2}} \right) \le \frac{1}{{{\bf{b}} - {\bf{a}}}}\int_{\bf{a}}^{\bf{b}} {f\left( {\bf{x}} \right)d{\bf{x}}} \le \frac{1}{{2^n }}\sum\limits_{\bf{c}} {f\left( {\bf{c}} \right)}, \end{align*} where $\sum\limits_{\bf{c}} {f\left( {\bf{c}} \right)} : = \sum\limits_{\mathop {c_i \in \left{ {a_i ,b_i } \right}}\limits_{1 \le i \le n} } {f\left( {c_1, c_2, \ldots ,c_n } \right)}$. Some other related result are given.