Inequalities for mixed $p$-affine surface area

Details Inequalities for mixed $p$-affine surface area Inequalities for mixed $p$-affine surface area

2008-12-24

arxiv.org/abs/0812.4550v1

Math. Ann. (2010) 347:703-737

Mathematics

Metric Geometry

39 pages

Scientific paper

10.1007/s00208-009-0453-2

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.

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