cell perimeter over area (area over volume in 3D) function

Is there an efficient way to obtain perimeter-over-area (in 2D) and area-over-volume (in 3D) functions for each cell?

In my DG formulation, I have an interior penalty term (Cip) that depends on maximum of Area-over-Volume (AoV) between the two elements that share a common edge (or face in 3D). So, I am looking for an AoV function that can be employed in the Cip definition below. My elements are simplex (triangle and tetrahedral).

# assume P is my polynomial order

def Max(a,b): return ( a+b + abs(a-b) ) / Constant(2.0)

dim = mesh.geometry().dim()

def Cip(dim, P, AoV):

    coef = (P+1)*(P+dim) / (2.0*dim)

    return coef * Max( AoV('+'), AoV('-') )

Regards,

Alireza