In this paper, we give a new higher dimensional Hermite-Hadamard inequality for a function $f:\prod\limits_{i=1}^n[a_i,b_i]\subset\mathbb{R}^n\rightarrow\mathbb{R}$ which is semiconvex of rate $(k_1,k_2,…,k_n)$ on the co-ordinates. This generalizes some existing results on Hermite-Hadamard inequalities of S.S. Dragomir. In addition, we explain the Hermite-Hadamard inequality from the point of view of optimal mass transportation with cost function $c(x,y):=f(y-x)+\sum_{i=1}^n\frac{k_i}{2}\big|x_i-y_i\big|^2$, where $f(\cdot):\prod\limits_{i=1}^n[a_i,b_i]\rightarrow[0,\infty)$ is semiconvex of rate $(k_1,k_2,…,k_n)$ on the co-ordinates and $x=(x_1,x_2,…,x_n)$, $y=(y_1,y_2,…,y_n)\in\prod\limits_{i=1}^n[a_i,b_i]$. Furthermore, by using the higher dimensional Hermite-Hadamard inequality, we compare the transport cost in different transport models on the sphere $\mathbb{S}^2$.