In this paper, we present a method to lift a homogeneous polynomial of $n$ variables and degree $d$ to a permutation symmetric tensor in $\otimes^dR^n$, which can be regarded as a variable grouped multiliear function with $n\times d$ variables, and map the vertices of a polyhedron in $R^n$ to a finite point set in $R^{n\times d}$,so to prove the positive definiteness of given homogeneous polynomials on the given polyhedron by calculating values of the constructed tensors on the set of finite points together with the barycenter partition technique.