The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang ), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz).
In order to achieve it, we investigate the graph functions of the given Lipschitz star body K: Along almost all directions u, we can define the graph functions on an open dense subset of the orthogonal projection of K onto u⊥.
General affine invariances related to Mahler volume are introduced. We establish their
affine isoperimetric inequalities by using a symmetrization scheme that involves a
total of 2n elaborately chosen Steiner symmetrizations at a time. The necessity of
this scheme, as opposed to the usual Steiner symmetrization, is demonstrated with an
example (see the Appendix).