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This thesis deals with the polar Orlicz-Minkowski problems and the p-capacitaryOrlicz-Petty bodies. The polar Orlicz-Minkowski problems are introduced and thesolvability of such problems is discussed under different conditions. In particular,under certain condition on φ; the existence of a solution is proved for a nonzero finite measure μ on Sn-1 which is not concentrated on any hemisphere of Sn-1: Theexistence of the p-capacitary Orlicz-Petty bodies is also established. The Orlicz andLq geominimal capacities with respect to K0 and S0 are proposed and their properties,such as invariance under orthogonal matrices, isoperimetric type inequalities andcyclic type inequalities are provided as well.

On the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies