In this paper, we give a new higher dimensional Hermite–Hadamard inequality for a function , which is semiconvex of rate (k1, k2, … , kn) on the coordinates. 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 , where is semiconvex of rate (k1, k2, … , kn) on the coordinates and , . Furthermore, by using the higher dimensional Hermite–Hadamard inequality, we compare the transport cost in different transport models on the sphere , relating our main result with transport models, which are related to the majorization, and as an interesting by‐product, we also reproof the Pólya–Szegö inequality.