Extreme properties of quermassintegrals of convex bodies

In this paper, we establish two theorems for the quermassintegrals of convex bodies, which are the generalizations of the well-known Aleksandrov's projection theorem and Loomis-Whitney's inequality, respectively. Applying these two theorems, we obtain a number of inequalities for the volumes of projections of convex bodies. Besides, we introduce the concept of the perturbation element of a convex body, and prove an extreme property of it.