log-minkowski inequalities for the lp $l_{p}$-mixed quermassintegrals

Abstract Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies. Recently, Stancu proved the log-Minkowski inequality for mixed volumes; Wang, Xu, and Zhou gave the Lp $L_{p}$ version of Stancu’s results. In this paper, we define the Lp $L_{p}$-mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the Lp $L_{p}$-mixed quermassintegrals. As its application, we establish the Lp $L_{p}$-mixed affine isoperimetric inequality. In addition, we also consider the dual log-Minkowski inequalities for the Lp $L_{p}$-dual mixed quermassintegrals.