Generalization of the Apollonius theorem for simplices and related problems

TL;DR

This paper extends the classical Apollonius theorem to m-simplices in higher-dimensional Euclidean spaces, providing new geometric properties and applications in optimization, shape analysis, and numerical methods.

Contribution

It introduces a generalized Apollonius theorem for m-simplices in n-dimensional space and applies it to problems in minimal enclosing surfaces, shape quality, and root-finding error analysis.

Findings

Derived a generalized Apollonius theorem for m-simplices.

Applied the theorem to minimal enclosing spherical surfaces.

Analyzed convergence and error estimates for root-finding on simplices.

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.