The Convex Hull of a Hypersurface
The Convex Hull of a Hypersurface
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超曲面的凸包
DOI：
10.1112/plms/s3-50.2.370
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发表时间：
1985
影响因子：
1.8
通讯作者：
M. C. Romero
中科院分区：
文献类型：
--
作者：
S. Robertson;M. C. Romero
设M是一个光滑的（即C）紧连通的无边界M流形，设/:M -> E是M在欧几里得（M + l）空间E中的光滑浸没。/(M)的凸包J^{f)是包含/(M)的E的所有凸子集的交集。它遵循出售Jif (f)是同胚的关闭(m + l)盘D + 1和它的边界 / / ( / ) 在E + 1 C ^超曲面的E \ CMiffeomorphic msphere s .我们进行这项工作的目的是描述的分解 / / ( / ) 到相关的子集的联系集的凸体的尺寸m + 1在£+ 1在滚动或休息时,在它的任何支持超平面。H(f)的这些子集称为面板，这样定义的H(f)分解称为面板结构。结果表明，取M = S并将注意力限制在光滑嵌入上并不会损失一般性。这是因为对于任何平滑浸入f:M-*E + 1，存在平滑嵌入g: S -+ E + 1，使得3#>{g) = Jt？(f)因此H(g) = H(f)我们证明了在E + l中S的光滑嵌入的空间${m)中存在残差子集，在该子集上面板结构“表现良好”。对于这种嵌套，面板与b[6]讨论的岩心分层地层密切相关。我们还得到了面板欧拉数之间的关系，这可以看作是多面体欧拉关系的推广。本文的当前版本包括了审稿人在形式和内容上建议的一些改进。
Let M be a smooth (that is, C) compact connected m-manifold without boundary, and let / : M -> E be a smooth immersion of M in Euclidean (m + l)-space E. The convex hull J^{f) of/(M) is the intersection of all the convex subsets of E that contain / (M). It follows that Jif(f) is homeomorphic to the closed (m+ l)-disc D + 1 and its frontier / / ( / ) in E + 1 is a C^hypersurface of E\ CMiffeomorphic to the msphere S. Our aim in this work is to describe a decomposition of / / ( / ) into subsets that are related to the 'contact sets' of a convex body of dimension m+ 1 in £ + 1 when rolling or resting on any of its supporting hyperplanes. These subsets of H(f) are called panels and the decomposition of H(f) so defined is called the panel structure. It turns out that there is no loss of generality in taking M = S and confining attention to smooth embeddings. This is because for any smooth immersion f:M-*E + l there is a smooth embedding g: S -+ E + 1 such that 3#>{g) = Jt?(f) and hence H(g) = H(f). We show that there is a residual subset of the space ${m) of smooth embeddings of S in E + l on which the panel structure is 'well-behaved'. For such embeddings the panels are closely related to the strata of the core stratification discussed in [6]. We also obtain a relation between the Euler numbers of the panels that may be regarded as a generalization of the Euler relation for polyhedra. The present version of this paper includes a number of improvements in both form and content suggested by the referee.