Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalization of the Apollonius theorem for simplices and related problems

Michael N. Vrahatis

TL;DR

The paper extends the Apollonius theorem from triangles to $m$-simplices in ${\mathbb{R}}^n$ ($n\ge m$), deriving a central identity for the $i$-th median in terms of edge lengths and the median length $\|\mu_i^m\|$ and providing explicit formulas for median lengths and related barycentric quantities. It then applies these results to core geometric problems, including minimal enclosing spheres via barycentric radii, and introduces simplex thickness as a quality metric through the barycentric inradius. The work surveys and extends fundamental theorems (Jung, Gale, Steinhagen) to simplices and various spaces, establishing bounds and exact values for circumradius, inradius, and width, and presents practical algorithmic tools such as simplex bisection with convergence guarantees. Overall, it unifies higher-dimensional simplex geometry with classical triangle results, yielding both theoretical insights and practical methods for geometric optimization and numerical computation.

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.

Generalization of the Apollonius theorem for simplices and related problems

TL;DR

The paper extends the Apollonius theorem from triangles to -simplices in (), deriving a central identity for the -th median in terms of edge lengths and the median length and providing explicit formulas for median lengths and related barycentric quantities. It then applies these results to core geometric problems, including minimal enclosing spheres via barycentric radii, and introduces simplex thickness as a quality metric through the barycentric inradius. The work surveys and extends fundamental theorems (Jung, Gale, Steinhagen) to simplices and various spaces, establishing bounds and exact values for circumradius, inradius, and width, and presents practical algorithmic tools such as simplex bisection with convergence guarantees. Overall, it unifies higher-dimensional simplex geometry with classical triangle results, yielding both theoretical insights and practical methods for geometric optimization and numerical computation.

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.
Paper Structure (12 sections, 38 theorems, 67 equations)

This paper contains 12 sections, 38 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Generalization of the Apollonius theorem for simplices
  3. Notation, definitions, and preliminary concepts
  4. Generalization of the Apollonius theorem
  5. Generalizations of median properties
  6. Applications of the generalized Apollonius theorem
  7. Simplex and set enclosing
  8. Jung's enclosing theorem and the minimum enclosing ball problem
  9. Generalizations of Jung's theorem to other spaces and geometries
  10. Simplex thickness
  11. Relations between the circumradius, inradius, diameter and width
  12. Simplex bisection

Key Result

Theorem 1.1

In any triangle $\Delta A B C$ with vertices $A$, $B$, $C$ and lengths of its sides $\ell (\,\,\overline{\!\!AB})$, $\ell (\,\,\overline{\!\!AC})$ and $\ell (\,\,\overline{\!\!BC})$, if $\overline{\!\!AD}$ is a median with length $\ell (\,\,\overline{\!\!AD})$, it holds that:

Theorems & Definitions (107)

  • Theorem 1.1: Apollonius' theorem
  • Definition 2.1
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 97 more