Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms

In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.

An Extension of Polynomial Integrability to Dual Quermassintegrals

A bounded domain $K$ 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 domains was given recently. Here we obtain a generalization of these results in the setting of dual quermassintegrals. We also address the associated smoothness issues.