We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these {\em covolume polynomials\/} and Lorentzian polynomials. While these are distinct notions, we prove that, like Lorentzian polynomials, covolume polynomials have M-convex support and generalize the notion of log-concave sequences. In fact, we prove that covolume polynomials are `sectional log-concave', that is, the coefficients of suitable restrictions of these polynomials form log-concave sequences. We observe that Chern classes of globally generated bundles give rise to covolume polynomials, and use this fact to prove that certain polynomials associated with {\em Segre classes\/} of subschemes of products of projective spaces are covolume polynomials. We conjecture that the same polynomials may be Lorentzian after a standard normalization operation. Finally, we obtain a combinatorial application of a particular case of our Segre class result. We prove that the {\em adjoint polynomial\/} of a convex polyhedral cone contained in the nonnegative orthant, and sharing a face with it, is a covolume polynomials. This implies that these adjoint polynomials are M-convex and sectional log-concave, and in fact Lorentzian after a suitable change of variables.