This is an announcement for the paper "Inequalities for mixed $p$-affine surface area" by Elisabeth Werner and Deping Ye. Abstract: 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. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A20, 53A15 Remarks: 39 pages The source file(s), MixedLp.tex: 97032 bytes, is(are) stored in gzipped form as 0812.4550.gz with size 26kb. The corresponding postcript file has gzipped size 162kb. Submitted from: elisabeth.werner@case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0812.4550 or http://arXiv.org/abs/0812.4550 or by email in unzipped form by transmitting an empty message with subject line uget 0812.4550 or in gzipped form by using subject line get 0812.4550 to: math@arXiv.org.