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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 $L_{p}$-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. [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]

We introduce the notion of dual mixed quermassintegral quotient functions and establish the Brunn-Minkowski inequalities for them in this paper. [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 notion of Lp-curvature image and proved an inequality for the volumes of convex body and its Lp-curvature image. In this paper, we first give an monotonic property of Lp-curvature image. Further, we establish two inequalities for the Lp-curvature image and its polar, respectively. Finally, an inequality for the volumes of Lp-projection body and Lp-curvature image is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we establish two cyclic Brunn–Minkowski inequalities for the general ith width-integrals and general ith chord-integrals, respectively. Our works bring the cyclic inequality and Brunn–Minkowski inequality together. [ABSTRACT FROM AUTHOR]

Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies. Recently, Stancu proved the log-Minkowski inequality for mixed volumes; Wang, Xu, and Zhou gave the L p version of Stancu's results. In this paper, we define the L p -mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the L p -mixed quermassintegrals. As its application, we establish the L p -mixed affine isoperimetric inequality. In addition, we also consider the dual log-Minkowski inequalities for the L p -dual mixed quermassintegrals. [ABSTRACT FROM AUTHOR]

Lutwak, Yang, and Zhang introduced the concept of (p,q)-mixed volume whose special cases contain the Lp-mixed volume and the Lp-dual mixed volume. In this article, associated with the (p,q)-mixed volumes, we establish related cyclic inequalities, monotonic inequalities, and product inequalities. [ABSTRACT FROM AUTHOR]