The notion of general mixed chord integrals of star bodies was introduced by Feng and Wang. In this paper, we extend the concept of the general mixed chord integrals to general L p -mixed chord integrals of star bodies. Based on this new notion, we study their extremum values and obtain an Aleksandrov-Frenchel type and a cyclic inequality for general L p -mixed chord integrals of star bodies, respectively. Further, as applications, we establish two Brunn-Minkowski type inequalities for L p -radial bodies. Finally, we get an interesting identical equality on combining L p -radial bodies.MSC: 52A20; 52A40; 52A39

Lutwak introduced the L p -harmonic radial body of a star body. In this paper, we define the notion of asymmetric L pharmonic radial bodies and study their properties. In particular, we obtain the extremum values of dual quermassintegrals and the volume of the polars of the asymmetric L p -harmonic radial bodies, respectively.