A characterization of dual quermassintegrals and the roots of dual steiner polynomials

For any $I\subset \mathbb{R}$ finite with $0\in I$, we provide a characterization of those tuples $({\omega }_i{)}_{i\in I}$ of positive numbers which are dual querma\ss integrals of two star bodies. It turns out that this problem is related to the moment problem. Based on this relation we also get new inequalities for the dual querma\ss integrals. Moreover, the above characterization will be the key tool in order to investigate structural properties of the set of roots of dual Steiner polynomials of star bodies.