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Matching centroids by a projective transformation

机译：通过投影变换匹配质心

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Abstract Given two subsets of Rddocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}^d$$end{document}, when does there exist a projective transformation that maps them to two sets with a common centroid? When is this transformation unique modulo affine transformations? We study these questions for 0- and d-dimensional sets, obtaining several existence and uniqueness results as well as examples of non-existence or non-uniqueness. If both sets have dimension 0, then the problem is related to the analytic center of a polytope and to polarity with respect to an algebraic set. If one set is a single point, and the other is a convex body, then it is equivalent by polar duality to the existence and uniqueness of the Santaló point. For a finite point set versus a ball, it generalizes the Möbius centering of edge-circumscribed convex polytopes and is related to the conformal barycenter of Douady-Earle. If both sets are d-dimensional, then we are led to define the Santaló point of a pair of convex bodies. We prove that the Santaló point of a pair exists and is unique, if one of the bodies is contained within the other and has Hilbert diameter less than a dimension-depending constant. The bound is sharp and is obtained by a box inside a cross-polytope.
机译：摘要 给定 Rddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}^d$$end{document}，什么时候存在将它们映射到两个具有公共质心的集合的投影变换？这种变换何时为唯一模仿射变换？我们研究了 0 维和 d 维集合的这些问题，获得了几个存在性和唯一性结果以及不存在或非唯一性的例子。如果两个集合的维数均为 0，则问题与多面体的解析中心和代数集的极性有关。如果一个集合是单点，而另一个集合是凸体，那么它就等价于圣塔洛点的存在性和唯一性。对于有限点集与球的比较，它推广了边缘外接凸多面体的莫比乌斯中心，并且与 Douady-Earle 的共形重心有关。如果两个集合都是 d 维的，那么我们就会定义一对凸体的 Santaló 点。我们证明一对的桑塔洛点存在并且是唯一的，如果其中一个物体包含在另一个物体中并且希尔伯特直径小于一个取决于维度的常数。边界是尖锐的，由交叉多面体内的盒子获得。

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