The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved Lp John ellipsoids. The analog of Ball’s volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.

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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 [36]), 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).

We establish a dual version of the Loomis-Whitney inequality for isotropic measures with complete equality conditions, where the sharp lower bound is given in terms of the volumes of hyperplane sections. For the special case of cross measures, we can drop the condition that the underlying body has centroid at the origin, yielding an independent proof of a result of Meyer's.