In this paper, we demonstrate the existence part of the discrete Orlicz Minkowski problem, which is a non-trivial extension of the discrete Lp Minkowski problem for 0 < p < 1.

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).

The conjecture about the Orlicz Pólya–Szegö principle posed in [42] is proved. The cases of equality are characterized in the affine Orlicz Pólya–Szegö principle with respect to Steiner symmetrization and Schwarz spherical symmetrization.

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The nth power of the volume functional Vnof polytopes P in R^n, according to dimensions of the spaces spanned by any nunit outer normal vectors of P, is decomposed into nhomogeneous polynomials of degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R^3 are established, which essentially characterize the geometric and algebraic structures of polytopes.

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson as a generalization of Thurston’s circle packing metric. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo proved the infinitesimal rigidity and then Luo proved the global rigidity. In this paper, based on an observation of Zhou, we prove this conjecture for inversive distance in (−1, +∞)by variational principles. We also study the global rigidity of a combinatorial curvature with respect to the inversive distance circle packing metrics where the inversive distance is in (−1, +∞).