A first step towards a dual Orlicz-Brunn-Minkowski theory for star sets was taken by Zhu, Zhou, and Xue \cite{ZZ, ZZX}. In this essentially independent work we provide a more general framework and results. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of B\"{o}r\"{o}czky, Lutwak, Yang, and Zhang. A new addition of star sets called radial $M$-addition is also introduced and shown to relate to the radial Orlicz addition.

The centro-affine Minkowski problem in affine differential geometry is considered. Existence for the solution of the discrete centro-affine Minkowski problem is proved.

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, +∞).