Sections of convex bodies
Sections of convex bodies
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DOI：
10.1353/ajm.1996.0021
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发表时间：
1996-04
影响因子：
1.7
通讯作者：
Gaoyong Zhang
中科院分区：
文献类型：
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作者：
Gaoyong Zhang
推广的Busemann-Petty问题问：如果[inline-graphic xmlns：xlink=”http://www.w3.org/1999/xlink“xlink：href=“01 i”/]中的中心对称凸体的i维中心截面的体积小于另一个这样的凸体的体积，那么凸体的体积是否也更小？证明了如果2是n，则答案是否定的。二维截面的情况仍然是开放的。证明使用的技术在功能分析和Radon变换格拉斯曼。它还需要i -相交体的概念，它推广了相交体的概念。证明了投影体、极投影体及其中心截面的体积不等式。它们与最大切片问题有关。
The generalized Busemann-Petty problem asks: If the volume of i -dimensional central section of a centrally symmetric convex body in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is smaller than that of another such body, is the volume of the body also smaller? It is proved that the answer is negative if 2 i n . The case of a 2-dimensional section remains open. The proof uses techniques in functional analysis and Radon transforms on Grassmannians. It also requires the notion of an i -intersection body which generalizes the notion of an intersection body. Inequalities among the volumes of projection bodies, polar projection bodies and their central sections are proved. They are related to the maximal slice problem.