A Form of Alexandrov-Fenchel Inequality

Abstract

There are many important integral formulae and integral inequalities for convex bodies (see [4], [18]). The Brunn-Minkowski inequality and Alexandrov-Fenchel inequality are among the most important integral inequalities in the theory of convex bodies, and the Minkowski type integral formulae and more general formulae of Chern are very useful in the global geometry of convex hypersurfaces. Most of these formulae and inequalities can be stated in the integral forms on the unit sphere Sn with the convexity assumption. It seems of interest to establish similar results without the convexity assumption. For a convex body, the polar of the body is also convex. The support function of the convex body corresponds to the gauge function of its polar body. In other words, the geometry of a convex body can be reflected from its polar dual. With this relation, we will introduce a class of domains in Rn+1 called k∗-convex (see Definition 4.2) as a natural generalization of convex bodies. We will derive a form of the Alexandrov-Fenchel