Surface bodies and p-affine surface area

Inequalities for mixed p - affine surface area ∗

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of Lp affine surface areas, mixed p-affine surface areas and other ...

On the p-affine surface area

Rényi Divergence and Lp-affine surface area for convex bodies

We show that the fundamental objects of the Lp-Brunn-Minkowski theory, namely the Lp-affine surface areas for a convex body, are closely related to information theory: they are exponentials of Rényi divergences of the cone measures of a convex body and its polar. We give geometric interpretations for all Rényi divergences Dα, not just for the previously treated special case of relative entropy ...

Affine surface area and convex bodies of elliptic type

If a convex body K in R is contained in a convex body L of elliptic type (a curvature image), then it is known that the affine surface area of K is not larger than the affine surface area of L. We prove that the affine surface areas of K and L can only be equal if K = L. 2010 Mathematics Subject Classification: primary 52A10; secondary 53A15

Width-integrals and Affine Surface Area of Convex 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.