L_p-dual three mixed quermassintegrals

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