A generalization of Hadamard’s inequality for convex functions

In this note, we give a counterexample to show that Hadamard’s inequality does not hold on a polyhedron in multi-dimensional Euclidean space. Then we give a sufficient condition on the polyhedron for Hadamard’s inequality to hold. Finally, we provide an approach to create a large class of polyhedra on which Hadamard’s inequality is true.