单纯形的阿波罗尼奥斯定理推广及相关问题
Generalization of the Apollonius theorem for simplices and related problems
AI总结：
本文将阿波罗尼奥斯定理推广至n维欧几里得空间的m-单纯形，基于与中线相关的结果，给出了求最小包围球面、单纯形厚度度量及寻根二分法收敛性和误差估计的应用。
摘要：
阿波罗尼奥斯定理根据三角形边长给出了其中线长度。本文给出了该定理的直接推广，针对n维欧几里得空间（n大于或等于m）中的m-单纯形得出了结果。基于此，给出了与三角形中线相关性质的推广。此外，还给出了广义阿波罗尼奥斯定理及与中线相关结果的应用，用于获取：(a)包围给定单纯形或有界集的最小球面，(b)单纯形的厚度，其为单纯形质量或形状优劣提供了一种度量，以及(c)在单纯形上应用寻根二分法的收敛性和误差估计。
英文摘要：
The Apollonius theorem gives the length of a median of a triangle in terms of the lengths of its sides. The straightforward generalization of this theorem obtained for m-simplices in the n-dimensional Euclidean space for n greater than or equal to m is given. Based on this, generalizations of properties related to the medians of a triangle are presented. In addition, applications of the generalized Apollonius' theorem and the related to the medians results, are given for obtaining: (a) the minimal spherical surface that encloses a given simplex or a given bounded set, (b) the thickness of a simplex that it provides a measure for the quality or how well shaped a simplex is, and (c) the convergence and error estimates of the root-finding bisection method applied on simplices.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
