Brunn-Minkowski inequalities for star duals of intersection bodies and two additions

In this paper,we first establish the dual Brunn-Minkowski inequality for the star duals for the Lp radial sum.Furthermore,we give some Brunn-Minkowski inequalities for the star duals of intersection bodies for the Lp radial sum and the Lp harmonic Blaschke sum.

凸体的L_p-径向线性组合

结合径向线性组合的定义给出了凸体的Lp-径向线性组合的定义,并探讨出了该组合下凸体的对偶混合体积之间的大小关系,重点研究了凸体的Lp-径向线性组合的平均宽度的相关性质,同时还给出了凸体的Lp-径向线性组合和它的极体的平均宽度的下界。在充分研究凸体的Lp-径向线性组合性质的基础上,得到凸体的Lp-径向线性组合和它的Firey线性组合之间的一些关系式。最后,在这些定理的支撑下,得到了关于凸体的Firey线性组合的平均宽度的下界的一般性结论,若K,L∈K0n实数p≥1,λ,μ≥0,λ+μ=1,则M((λK*+pμL*)+(λK+pμL))≥4,等号成立当且仅当K与L都为单位球时。

On Isolations for the Brunn-Minkowski Inequality of Quermassintegrals and Dual Quermassintegrals

Lutwak proved the Brunn-Minkowski inequality for the quermassintegrals of Fiery Lρ-combination. Wang and Leng gave the Brunn-Minkowski inequality for the dual quermassintegrals of Lρ-harmonic radial combination. In the paper, we establish the isolate forms of the Brunn-Minkowski inequality for quermassintegrals and dual quermassintegrals,respectively.

Inequalities for L_p Mixed Intersection Bodies

Haberl and Ludwig defined the notions of L_p intersection bodies. In this paper,we introduce the L_p mixed intersection bodies, and establish some geometric inequalities for L_p mixed intersection bodies. Furthermore, the Busemann-Petty type problem for L_p mixed intersection bodies are shown.

Some Inequalities for L_p-Dual Mixed Quermassintegrals

Wang and Zhang defined a type of Lp-dual mixed quermassintegrals based on the Lp-radial combinations and dual quermassintegrals of star bodies. In the article, the product inequalities for this Lp-dual mixed quermassintegrals are established. As the applications, we obtain the lower bounds of dual quermassintegrals product. Further, the Brunn-Minkowski type inequality and the cycle inequality for the Lp-dual mixed quermassintegrals are given.

Lp径向Blaschke-Minkowski同态的Lp对偶几何表面积

2006年,Schuster提出了径向Blaschke-Minkowski同态的概念.随后,汪卫等人将其推广到Lp径向Blaschke-Minkowski同态.本文结合Lp对偶几何表面积,建立了Lp径向Blaschke-Minkowski同态的若干不等式,包括Brunn-Minkowski型不等式和单调不等式.并给出了Lp径向Blaschke-Minkowski同态的Busemann-Petty问题的肯定和否定形式.

Generalized Intersection Bodies with Parameter

In 2005, the classical intersection bodies and L_p intersection bodies were extended. Afterwards, the concept of gen-eral L_p intersection bodies and the generalized intersection bodies was introduced. In this paper, we define the generalized bodies with parameter. Besides, we establish the extremal values for volume, Brunn-Minkowski type inequality for radial combination and L_p harmonic Blaschke combination of this notion.