On mother bodies of convex polyhedra

If Omega is a bounded domain in R-N provided with a mass distribution rho(Omega) (e.g., Lebesgue measure restricted to Omega), another mass distribution mu sitting in Omega and producing the same external Newtonian potential as rho(Omega) is sometimes called a mother body of Omega, provided it is maximally concentrated in some sense. We first discuss the meaning of this and formulate five desirable properties ("axioms") of mother bodies. Then we show that convex polyhedra do have unique mother bodies in that sense made precise in the case that rho(Omega) is either Lebesgue measure on Omega, hypersurface measure on partial derivative Omega, or any mixture of these two.