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

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