In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized $p$-capacitary $L_q$ Minkowski problems with $q>1$ for both discrete and general measures.

For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

In this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke– Santal´o type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

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This paper deals with Lp geominimal surface area and its extension to Lp mixed geominimal surface area. We give an integral formula of Lp geominimal surface area by the p-Petty body and introduce the concept of Lp mixed geominimal surface area which is a natural extension of Lp geominimal surface area. Some inequalities, such as, analogues of Alexandrov–Fenchel inequalities, Blaschke–Santaló inequalities, and affine isoperimetric inequalities for Lp mixed geominimal surface areas are obtained.