An extension of polynomial integrability to dual quermassintegrals

A body $K$ is called polynomially integrable if its parallel section function $V_{n-1}(K\cap\{\xi^\perp+t\xi\})$ is a polynomial of $t$ (on its support) for every $\xi$. A complete characterization of such bodies was given recently. Here we obtain a generalization of these results in the setting of dual quermassintegrals. We also address the associated smoothness issues.