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Motivated by the celebrated work of Lutwak, Yang and Zhang [1] on (p,q)\left(p,q)-mixed volumes and that of Feng and He [2] on (p,q)\left(p,q)-mixed geominimal surface areas, we in the present paper establish and confirm the affine isoperimetric and Brunn-Minkowski-type inequalities with respect to (p,q)\left(p,q)-mixed geominimal surface areas.

In this paper, we introduce the L p q-torsional measure for p ∈ ℝ and q > 1 by the L p variational formula for the q-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the L p Minkowski problem with respect to the q-torsional rigidity for discrete measures and general measures when 0 < p < 1 and q > 1 . [ABSTRACT FROM AUTHOR]

In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.

In 2006, Schuster introduced the concept of Blaschke-Minkowski homo-morphism of convex bodies. In this paper, we introduce the Lp-mixed Blaschke-Minkowski homomorphism in Lp-Brunn-Minkowski theory. We then further study the Shephard type problem involving an affirmative answer and two negative answers for Lp-mixed Blaschke-Minkowski homomorphism. [ABSTRACT FROM AUTHOR]

The notion of (p, q)-mixed volume was first introduced by Lutwak, Yang and Zhang in 2018. According to this concept, Li, Wang and Zhou defined the (p, q)- mixed affine surface areas in 2019. In this paper, we establish some inequalities including cyclic inequality, monotonous inequality and product inequality for the (p, q)-mixed affine surface areas. [ABSTRACT FROM AUTHOR]

The notion of the mixed brightness was first introduced by Lutwak. Recently, Abardia and Bernig presented complex projection bodies. Based on this notion, we define the mixed complex brightness integrals and establish related Aleksandrov–Fenchel inequality, cyclic inequality and monotonicity inequality, respectively. [ABSTRACT FROM AUTHOR]

Based on the notion of Lp-mean zonoids which is introduced by Xi, Guo and Leng, we define the notion of asymmetric Lp-mean zonoids and establish its an affine inequality. Further, we give the extremal values for volume, quermassintegrals of this notion and dual quermassintegrals of polar of this notion. Finally, associated with p-radial mean bodies, two monotonic inequalities for the asymmetric Lp-mean zonoids are obtained. [ABSTRACT FROM AUTHOR]

The notion of radial Blaschke-Minkowski homomorphisms was presented by Schuster. Afterwards, Wang et al. introduced Lp radial Blaschke-Minkowski homomorphisms. In this paper, associated with Lp-dual affine surface areas, we establish some inequalities including the Brunn-Minkowski type inequality, cyclic inequality and monotonic inequalities, and give an affirmative answer and a negative answer of Busemann-Petty problem for the Lp radial Blaschke-Minkowski homomorphisms. [ABSTRACT FROM AUTHOR]

Schuster introduced radial Blaschke-Minkowski homomorphisms. Recently, they were generalized to $L_{p}$ radial Blaschke-Minkowski homomorphisms by Wang et al. In this paper, we first establish Brunn-Minkowski type inequalities for some $L_{q}$ radial sums of $L_{p}$ radial Blaschke-Minkowski homomorphisms. Further, we consider monotonic inequalities for $L_{p}$ radial Blaschke-Minkowski homomorphisms. [ABSTRACT FROM AUTHOR]

Following the discovery of general $L_{p}$-projection bodies by Ludwig, Haberl and Schuster determined the extremum of the volume of the polars of this family of $L_{p}$-projection bodies. In this paper, the result of Haberl and Schuster is extended to all dual quermassintegrals, and a dual counterpart for the quermassintegrals of general $L_{p}$-projection bodies is also obtained. Moreover, the extremum of the $L_{q}$-dual affine surface areas of polars of general $L_{p}$-projection bodies are determined. [ABSTRACT FROM AUTHOR]

The notion of general mixed chord integrals of star bodies was introduced by Feng and Wang. In this paper, we extend the concept of the general mixed chord integrals to general $L_{p}$-mixed chord integrals of star bodies. Based on this new notion, we study their extremum values and obtain an Aleksandrov-Frenchel type and a cyclic inequality for general $L_{p}$-mixed chord integrals of star bodies, respectively. Further, as applications, we establish two Brunn-Minkowski type inequalities for $L_{p}$-radial bodies. Finally, we get an interesting identical equality on combining $L_{p}$-radial bodies. [ABSTRACT FROM AUTHOR]

Lutwak and Zhang proposed the concept of $L_{p}$-centroid bodies. Then Feng and Wang gave the notion of general $L_{p}$-centroid bodies. In this article, based on the $L_{p}$-dual affine surface area, we address Shephard type problems for the general $L_{p}$-centroid bodies. [ABSTRACT FROM AUTHOR]

Lutwak defined the dual Blaschke combination of star bodies. In this paper, based on the $L_{p}$-dual Blaschke combination of star bodies, we define the general $L_{p}$-dual Blaschke bodies and obtain the extremal values of their volume and $L_{p}$-dual affine surface area. Further, as the applications, we study two negative forms of the $L_{p}$-Busemann-Petty problems. [ABSTRACT FROM AUTHOR]

The notion of mixed-brightness integrals was introduced by Li and Zhu. In this paper, motivated by the notion of general $L_{p}$-projection bodies, introduced by Haberl and Schuster, we define general $L_{p}$-mixed-brightness integrals and determine their extremal values, as well as several other inequalities for them. [ABSTRACT FROM AUTHOR]

Lutwak introduced the harmonic Blaschke combination and the harmonic Blaschke body of a star body. Further, Feng and Wang introduced the concept of the L-harmonic Blaschke body of a star body. In this paper, we define the notion of general L-harmonic Blaschke bodies and establish some of its properties. In particular, we obtain the extreme values concerning the volume and the L-dual geominimal surface area of this new notion. [ABSTRACT FROM AUTHOR]