Inequalities Relating to Lp-Version of the Petty’s Conjectured Projection Inequality
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摘要: 在凸体理论中,投影不等式的Petty猜想是一个著名的公开问题．首先通过利用Lp-混合体积和Lp-对偶混合体积的概念、Lp-投影体和几何体Γ-pK的关系、Bourgain-Milman不等式和Lp-Busemann-Petty不等式，建立了一个联系投影不等式Petty猜想的Lp-形式的不等式．继而对于每一个关于原点对称的凸体,应用Jensen不等式和几何体Γ-pK的单调性,分别给出了投影不等式Petty猜想的Lp-形式的一个逆向不等式和Lp-Petty投影不等式的一个逆向不等式．
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关键词:
- Lp-形式 /
- Petty投影不等式 /
- 投影不等式的Petty猜想：Lp-投影体 /
- 逆向不等式
Abstract: Petty's conjectured projection inequality is a famous open problem in convex bodies theory.It was shown that an inequality relating to Lp-version of the Petty's conjectured projection inequality by using the notions of the Lp-mixed volume and the Lp-dual mixed volume,the relation of the Lp-projection body and the geometric body Г-pK,the Bourgain-Milman inequality and the Lp-Busemann-Petty inequality.In addition,for each origin-symmetric convex body,applying the Jensen inequality and the monotonicity of the geometric body Г-pK,the reverses of Lp-version of the Petty's conjectured projection inequality and the Lp-Petty projection inequality were given respectively. -
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