λ-相交体的Busemann-Petty问题

马统一

数学学报 ›› 2013, Vol. 56 ›› Issue (2) : 263-278.

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数学学报 ›› 2013, Vol. 56 ›› Issue (2) : 263-278. DOI: 10.12386/A2013sxxb0028
论文

λ-相交体的Busemann-Petty问题

    马统一
作者信息 +

On the Busemann-Petty Problem of λ-Intersection Bodies

    Tong Yi MA
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文章历史 +

摘要

对于λ<nλ≠0,Rubin引进了Rn中原点中心对称星体Kλ-相交体IBλ(K)的概念. 本文研究IBλ(K)⊆IBλ(L)是否必定⇒voln(K)≤voln(L)(或voln(K)≥voln(L))的Busemann-Petty问题. 其结论概括为:λ-相交体的Busemann-Petty问题的解为肯定当且仅当Rn中任意一个原点中心对称星体都是一个λ-相交体.这些结果推广了经典相交体的Busemann-Petty问题的特定正解.

Abstract

For λ<n and λ=0, Rubin introduced the concept of the λ-intersection body IBλ(K) of an origin-symmetric star body K in Rn. In this paper, we consider Busemann-Petty's problem of whether IBλ(K)⊆IBλ(L) implies voln(K)≤voln(L) (or voln(K)≥voln(L)). We proved that Busemann-Petty's problem of λ-intersection body in Rn has a positive answer if and only if any origin-symmetric star body in Rn is a λ-intersection body. Our results generalize the specific affirmative answer of classic intersection bodies to Busemann-Petty problem.

关键词

凸体 / 星体 / λ-相交体 / 等距嵌入到Lp / 广义余弦变换

Key words

convex body / star body / λ-intersection bodies / isometrically embedded in Lp / generalized cosine transforms

引用本文

导出引用
马统一. λ-相交体的Busemann-Petty问题. 数学学报, 2013, 56(2): 263-278 https://doi.org/10.12386/A2013sxxb0028
Tong Yi MA. On the Busemann-Petty Problem of λ-Intersection Bodies. Acta Mathematica Sinica, Chinese Series, 2013, 56(2): 263-278 https://doi.org/10.12386/A2013sxxb0028

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基金

国家自然科学基金资助项目(11161019);甘肃省教育厅研究生导师科研基金资助项目(1009B-09)

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