Lp -dual three mixed quermassintegrals

In the paper, the concept of Lp -dual three-mixed quermassintegrals is introduced. The formula for the Lp -dual three-mixed quermassintegrals with respect to the p-radial addition is proved. Inequalities of Lp -Minkowski, and Brunn-Minkowski type for the Lp -dual three-mixed quermassintegrals are established. The new Lp -Minkowski inequality is obtained that generalize a family of Minkowski type inequalities. The Lp -Brunn-Minkowski inequality is used to obtain a series of Brunn-Minkowski type inequalities.