For 0 < p < 1, Haberl and Ludwig defined the notions of symmetric and asymmetric L p -intersection bodies. Recently, Wang and Li introduced the general L p -intersection bodies. In this paper, we give the L p -dual geominimal surface area forms for the extremum values and Brunn-Minkowski type inequality of general L p -intersection bodies. Further, combining with the L p -dual geominimal surface areas, we consider Busemann-Petty type problem for general L p -intersection bodies.

Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered the Busemann-Petty problem for volume forms. Whereafter, Wang, Liu and He presented the L p radial Blaschke-Minkowski homomorphisms and extended Schuster’s results. In this paper, associated with L p dual affine surface areas, we give an affirmative and a negative form of the Busemann-Petty problem and establish two Brunn-Minkowski inequalities for the L p radial Blaschke-Minkowski homomorphisms.