In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.

in this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. following this, we establish the minkowski and brunn-minkowski inequalities for volumes difference function of the projection and intersection bodies.

In this paper, applying for the Minkowski?s and H?lder?s integral inequalities, we obtain four theorems about (p, q)-mixed volume involving Lp centroid bodies intersection bodies, respectively. The former two reveal convexity of functionals related to volume, in terms dual Blaschke addition introduced [Journal Geometric Analysis, 30 (2020) 3026-3034], latter expose monotonicity other volume.