Some Inequalities for \(p\)-Quermassintegrals

Abstract

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.