Let C-d,C-n be the convex cone consisting of real n-variate degree d forms that are strictly positive on R-n {0}. We prove that the Lebesgue volume of the sublevel set {g <= 1} of g is an element of C-d,n is a completely monotone function on C-d,C-n and investigate the related properties. Furthermore, we provide (partial) characterization of forms whose sublevel sets have finite Lebesgue volume.
