Extended affine surface area and zonotopes (Q1959058)

The authors introduce the notion of \(L_p\)-polar curvature image and provide an extensive study of its properties. They consider the family of sets \(\mathcal W^n\) as those convex bodies whose positive continuous curvature functions are the support functions of a zonoid. They also define the corresponding family \(\mathcal W^n_p\) for the \(L_p\)-theory. They show that there exists an equivalent relationship between the extended mixed affine surface area, of an arbitrary convex body and a body in \(\mathcal W^n\) (or \(\mathcal W^n_p\)), and the mixed volume, of an arbitrary convex body and a line segment. This approach is closely related to the properties of zonotopes, and with this approach most of the problems involving the area of projections of convex bodies will be remarkably promoted. In 1990 Lutwak proved that if \(K\) is an \(n\)-dimensional convex body and \(E\) is an ellipsoid, then if the areas of the projections of \(K\) do not exceed those of \(E\), it follows that \(\Omega (K) \leq \Omega (E)\), where \(\Omega (.)\) denotes the affine surface area. In this paper the authors use the equivalence between mixed volumes and extended affine surface area to reprove Lutwak's result. They also prove the \(L_p\)-version of this result. They also provide interesting applications of this approach to establish Aleksandrov's projection theorem and Petty-Schneider theorem for \(L_p\)-mixed quermassintegral version.