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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.

In this paper, we establish an inequality for the volumes of K and its radial p-th mean body R K by L-dual mixed volume. Further, the Shephard-type problem and a monotonitic inequality of R K are shown when p>0. [ABSTRACT FROM AUTHOR]