The present paper is devoted to study of the space of all weakly additive, order-preserving, normalized and positively-homogeneous functionals on a metric compactum. We construct an analogue of the modified Kantorovich--Rubinstein metric on the space \(OH(X)\) of all weakly additive, order-preserving, normalized and positively-homogeneous functionals on a metric compactum \(X.\) We investigate under what conditions subfunctors of the functor \(OH\) will be perfectly metrizable. We prove that under natural assumptions on \(X\) the triple \((\mathcal{F}^\omega_+(X), \mathcal{F}^{++}_+(X), \mathcal{F}^+_+(X))\) is homeomorphic to the triple \((Q,s, \textrm{rint}\, Q),\) where \(\mathcal{F}\) is a convex subfunctor of the functor \(OH_+.\)