Generalization of the Apollonius theorem for simplices and related problems
Abstract
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.
Keywords
Cite
@article{arxiv.2401.03232,
title = {Generalization of the Apollonius theorem for simplices and related problems},
author = {Michael N. Vrahatis},
journal= {arXiv preprint arXiv:2401.03232},
year = {2024}
}