Towards the Orlicz-Brunn-Minkowski theory for geominimal surface areas and capacity

This thesis is dedicated to study Orlicz-Petty bodies, the p-capacitary Orlicz- Brunn-Minkowski theory and the general p-affine capacity as well as isocapacitary inequalities. In the second chapter, the homogeneous Orlicz affine and geominimal surface areas are defined and their basic properties are established including homogeneity, affine invariance and continuity. Some related affine isoperimetric inequalities are proved. Similar results for the nonhomogeneous ones are proved as well. In the third chapter, we develop the p-capacitary Orlicz-Brunn-Minkowski theory by combining the p-capacity for p 2 (1; n) with the Orlicz addition of convex domains. In particular, Orlicz-Brunn-Minkowski type and Orlicz-Minkowski type inequalities are proved. In the last chapter, the general p-affine capacity for p 2 [1; n) is defined and its properties are discussed. Furthermore, the newly proposed general p-affine capacity is compared with many classical geometric quantities, e.g., the volume, the p-variational capacity and the p-integral affine surface area. Consequently, several sharp geometric inequalities for the general p-affine capacity are obtained. Theses inequalities extend and strengthen many well-known (affine) isoperimetric and (affine) isocapacitary inequalities