In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.

The main purposes of this paper are to establish some new Brunn– Minkowski inequalities for width-integrals of mixed projection bodies and affine surface area of mixed bodies, together with their inverse forms.

in this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. following this, we establish the minkowski and brunn-minkowski inequalities for volumes difference function of the projection and intersection bodies.

In this paper we establish the Lp-Minkowski inequality and Lp-Aleksandrov-Fenchel type inequality for Lp-dual mixed volumes of star duality of mixed intersection bodies, respectively. As applications, we get some related results. The paper new contributions that illustrate this duality of projection and intersection bodies will be presented. M.S.C. 2000: 52A40.

In this paper we establish the Aleksandrov-Fenchel type inequality for volume differences function of convex bodies and the Aleksandrov-Fenchel inequality for Quermassintegral differences of mixed projection bodies, respectively. As applications, we give positive solutions of two open problems. M.S.C. 2000: 52A40.

In this paper, the mixed Lp-surface area measures are defined and Lp Minkowski inequality is obtained consequently. Furthermore, projection for bodies established.

For n ≥ 2 a construction is given for convex bodies K and L in R n such that the orthogonal projection Ku can be translated inside Lu for every direction u, while the volumes of K and L satisfy Vn(K) > Vn(L). A more general construction is then given for n-dimensional convex bodies K and L such that the orthogonal projection Kξ can be translated inside Lξ for every k-dimensional subspace ξ of R...

Let Π be the projection operator, which maps every polytope to its projection body. It is well known that Π maps the set of polytopes, P, in R into P, that it is a valuation, and that for every P ∈ P, ΠP is affinely associated to P . It is shown that these properties characterize the projection operator Π. This proves a conjecture by Lutwak. Let Kn denote the set of convex bodies (i.e., of comp...