The answer to the following questions is nowadays standard: under which conditions is a given function the Gauss curvature or the mean curvature of a convex surface?

In its non-smooth version, these questions are known as Minkowski problem and Christoffel problem, respectively, which have been fully solved decades ago. The same question for other elementary symmetric functions of (generalized) principal curvatures is known as the Christoffel-Minkowski problem. Not much is known on its complete solution, except the breakthrough by Guan and Ma in the early 2000s, where sufficient conditions for the existence of solutions are provided under suitable regularity. A previous positive result was obtained by Firey in the 70s for bodies of revolution under suitable regularity. In a recent substantial advancement, Brauner, Hofstätter, and Ortega-Moreno presented a complete solution for Firey's setting without any regularity assumption, employing modern develpments in integral geometry and valuation theory. In this talk, we present an alternative solution we obtained with Mussnig. The methods build on basic machinery, drawing from the theory of Monge-Ampère equations, and allow us to recover explicit solutions.