Lutwak introduced the L p -difference body of a convex body as the Firey L p -combination of the body and its reflection at the origin. In this paper, we define the notion of asymmetric L p -difference bodies and study some of their properties. In particular, we determine the extremal values of the volumes of asymmetric L p -difference bodies and their polars, respectively.

The L p -geominimal surface area was introduced by Lutwak in 1996, which extended the important concept of the geominimal surface area. Recently, Wang and Qi defined the p-dual geominimal surface area, which belongs to the dual Brunn-Minkowski theory. In this paper, based on the concept of the dual Orlicz mixed volume, we extend the dual geominimal surface area to the Orlicz version and give its properties. In addition, the isoperimetric inequality, a Blaschke-Santaló type inequality, and the monotonicity inequality for the dual Orlicz geominimal surface areas are established. MSC: 52A30; 52A40

This paper devotes to establish complex dual Brunn-Minkowski theory. At first, we introduce the concepts of complex radial combination and complex radial-Blaschke combination, and obtain the relations between those two combinations and dual mixed volumes. Then, we extend the properties of real intersection body to the complex case. Finally, we prove some complex geometric inequalities about complex intersection bodies and complex mixed intersection bodies, such as dual Brunn-Minkowski type, dual Aleksandrov-Fenchel type and dual Minkowski type inequality. Moreover, as applications, we get some corollaries including an isoperimetric type inequality and a uniqueness theorem.

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Giannopoulos proved that a smooth convex body has minimal mean width position if and only if the measure ℎ ( ) (d ), supported on −1 , is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal -mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal -mean width of convex bodies and prove the existence and uniqueness of the minimal -mean width in its SL( ) images. In addition, we establish a characterization of the minimal -mean width, conclude the average ( ) with a variation of the minimal -mean width position, and give the condition for the minimum position of ( ).