Median hyperplanes in normed spaces -: a survey
Median hyperplanes in normed spaces -: a survey
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DOI：
10.1016/s0166-218x(98)00103-6
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发表时间：
1998-12-01
影响因子：
1.1
通讯作者：
Schöbel, A
中科院分区：
文献类型：
--
作者：
Martini, H;Schöbel, A
在这篇综述中，我们讨论了n维赋范空间中超平面的位置，即，我们给出了所有已知的结果和Minkowski空间中所谓的中值超平面问题的统一方法。我们描述了如何找到一个超平面H来最小化到给定的有限需求点集的距离的加权和f(H)。在稳健统计和运筹学中，这样的最优超平面称为中值超平面。在总结了欧几里得和矩形情形的已知结果之后，我们证明了对于所有由范数得到的距离度量d，其中一个极小化f(H)的超平面是所需点的n的仿射壳，而且每个中值超平面相对于给定的点集是一个平分的超平面。并给出了寻找固定斜率最优超平面的范数无关性。此外，我们还讨论了这些几何准则如何用于中值超平面的算法方法，并特别讨论了多面体范数的情况。最后给出了中值超平面的锐化关联准则对所有光滑范数的刻画。(C)1998 Elsevier Science B.V.保留所有权利。
In this survey we deal with the location of hyperplanes in n-dimensional normed spaces, i.e., we present all known results and a unifying approach to the so-called median hyperplane problem in Minkowski spaces. We describe how to find a hyperplane H minimizing the weighted sum f(H) of distances to a given, finite set of demand points. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. After summarizing the known results for the Euclidean and rectangular situation, we show that for all distance measures d derived from norms one of the hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median hyperplane is a halving one tin a sense defined below) with respect to the given point set. Also an independence of norm result for finding optimal hyperplanes with fixed slope will be given. Furthermore, we discuss how these geometric criteria can be used for algorithmical approaches to median hyperplanes, with an extra discussion for the case of polyhedral norms. And finally a characterization of all smooth norms by a sharpened incidence criterion for median hyperplanes is mentioned. (C) 1998 Elsevier Science B.V. All rights reserved.