Geominimal Surface Area Research Articles

Orlicz geominimal surface areas

In 1996, E. Lutwak extended the important concept of geominimal surface area to Lp version, which serves as a bridge connecting a number of areas of geometry: affine differential geometry, relative differential geometry, and Minkowskian geometry. In this paper, by using the concept of Orlicz mixed volume, we extend geominimal surface area to the Orlicz version and give some properties and an isoperimetric inequalities for the Orlicz geominimal surface areas. Mathematics subject classification (2010): 52A39, 52A40.

The L p -dual mixed geominimal surface area for multiple star bodies

According to the notion of the L p -mixed geominimal surface area of multiple convex bodies which were introduced by Ye et al., we define the concept of the L p -dual mixed geominimal surface area for multiple star bodies, and we establish several inequalities related to this concept.MSC:52A20, 52A40.

Formula omitted] mixed geominimal surface area

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.

Lp Geominimal Surface Areas and their Inequalities

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the $L_p$ geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santal\'{o} style inequality. Cyclic inequalities are established to obtain the monotonicity of the $L_p$ geominimal surface areas. Comparison between the $L_p$ geominimal surface area and the $p$-surface area is also provided.

General L p -harmonic Blaschke bodies

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

Some inequalities for geominimal surface areas

The concept of geominimal surface area is first introduced, and then, the affine surface area projection inequality is given by Petty. In recent years, associated with geominimal surface area, Lutwak obtained some significant results containing the extended affine projection inequality. In this paper, combining with curvature image and combinations of convex bodies, we get some inequalities for geominimal surface areas. Furthermore, the integral form of geominimal surface area is obtained.

The mixed $L_p$ geominimal surface areas for multiple convex bodies

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are proved. Related inequalities, such as, Alexander-Fenchel type inequality, Santal\'{o} style inequality, affine isoperimetric inequalities, and cyclic inequalities are established. Moreover, we also study some properties and inequalities for the $i$-th mixed $L_p$ geominimal surface areas for two convex bodies.

Lp-DUAL MIXED GEOMINIMAL SURFACE AREA

AbstractLutwak (Adv. Math., vol. 118(2), 1996, pp. 244–294) defined the notion of Lp-geominimal surface area based on Lp-mixed volumes. Recently, Wang and Qi (J. Inequal. Appl., vol. 2011, 2011, pp. 1–10) introduced the concept of Lp-dual geominimal surface area based on Lp-dual mixed volumes. In this paper, based on Lp-dual mixed quermassintegrals, we define the concept of Lp-dual mixed geominimal surface area and establish several inequalities for this new notion.

A GENERAL $L_p$-VERSION OF PETTY'S AFFINE PROJECTION INEQUALITY

About a decade ago Lutwak, Yang, and Zhang introduced the notion of $L_p$-projection body. More recently, Wang and Leng established an $L_p$-version of Petty's affine projection inequality. At the same time Ludwig discovered a family of general $L_p$-projection bodies and Haberl and Schuster established Petty's projection inequality for general $L_p$-projection bodies. In this paper we establish a general $L_p$-version of Petty's affine projection inequality for general $L_p$-projection bodies. Moreover, we obtain an analogous inequality for $L_p$-geominimal surface area.

L p -Dual geominimal surface area

Lutwak proposed the notion of L p -geominimal surface area according to the L p -mixed volume. In this article, associated with the L p -dual mixed volume, we introduce the L p -dual geominimal surface area and prove some inequalities for this notion. 2000 Mathematics Subject Classification: 52A20 52A40.