Some inequalities for the (p,q)-mixed geominimal surface areas and Lp radial Blaschke-Minkowski homomorphisms

Wang et al. introduced Lp radial Blaschke-Minkowski homomorphisms based on Schuster?s radial Blaschke-Minkowski homomorphisms. In 2018, Feng and He gave the concept of (p,q)-mixed geominimal surface area according to the Lutwak, Yang and Zhang?s (p,q)-mixed volume. In this article, associated with the (p,q)-mixed geominimal surface areas and the Lp radial Blaschke-Minkowski homomorphisms, we establish some inequalities including two Brunn-Minkowski type inequalities, a cyclic inequality and two monotonic inequalities.

Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the Lp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.

The General Dual Orlicz Geominimal Surface Area

In this paper, we study the general dual Orlicz geominimal surface area by the general dual Orlicz mixed volume which was introduced by Gardner et al. (2019). We find the conditions to the existence of the general dual Orlicz-Petty body and hence prove the continuity of the general geominimal surface area in the Orlicz setting (2010 Mathematics Subject Classification: 52A20, 53A15).

Dual Orlicz mixed geominimal surface areas

The notion of Lp-geominimal surface area was originally introduced by Lutwak in 1996. Recently, Feng andWang introduced the concept of Lp-dual mixed geominimal surface area based on Lp-dual mixed quermassintegrals. In this paper, based on dual Orlicz mixed quermassintegrals, we define the concept of dual Orlicz mixed geominimal surface area and establish some related inequalities for this new notion.

Busemann-Petty Problems for Quasi Lp Intersection Bodies

We investigate the Lp dual geominimal surface area and volume forms of Busemann-Petty problems for the quasi Lp intersection bodies and establish some new geometric inequalities. Our results provide a significant complement to the researches on Busemann-Petty problems for intersection bodies.

DualLp-Mixed Geominimal Surface Area and Related Inequalities

The integral formula of dualLp-geominimal surface area is given and the concept of dualLp-geominimal surface area is extended to dualLp-mixed geominimal surface area. Properties for the dualLp-mixed geominimal surface areas are established. Some inequalities, such as analogues of Alexandrov-Fenchel inequalities, Blaschke-Santaló inequalities, and affine isoperimetric inequalities for dualLp-mixed geominimal surface areas, are also obtained.