  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 $L_p$ affine surface areas, mixed
$p$-affine surface areas and other functionals via these bodies. The surprising
new element is that not necessarily convex bodies provide the tool for these
interpretations.

Metric Geometry;Functional Analysis