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 \(L_{p}\) version of Stancu’s results. In this paper, we define the \(L_{p}\)-mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the \(L_{p}\)-mixed quermassintegrals. As its application, we establish the \(L_{p}\)-mixed affine isoperimetric inequality. In addition, we also consider the dual log-Minkowski inequalities for the \(L_{p}\)-dual mixed quermassintegrals.