Some inequalities for $p$-quermassintegrals

In this paper, we generalize the notions of quermassintegrals, harmonic quermassintegrals, and affine quermassintegrals to $p$-quermassintegrals so that the cases $p=1, -1, -n$ of $p$-quermassintegrals are quermassintegrals, harmonic quermassintegrals, and affine quermassintegrals, respectively. Further, we obtain some inequalities associated with $p$-quermassintegrals, including $L_q$ Brunn–Minkowski-type inequalities, a monotonic inequality, and a Bourgain–Milman-type inequality.