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Towards the methodology for solving the minimum enclosing ball and related problems

Michael N. Vrahatis

Abstract

Methodology is provided towards the solution of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean space. Mathematical formulation and typical methods for solving this problem are presented. Also, the paper is focused on areas that are related to this problem, namely: (a) promise problems and property testing, (b) theorems for partitioning and enclosing (covering) a set, and (c) computation of the diameter of a set.

Towards the methodology for solving the minimum enclosing ball and related problems

Abstract

Methodology is provided towards the solution of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean space. Mathematical formulation and typical methods for solving this problem are presented. Also, the paper is focused on areas that are related to this problem, namely: (a) promise problems and property testing, (b) theorems for partitioning and enclosing (covering) a set, and (c) computation of the diameter of a set.

Paper Structure

This paper contains 6 sections, 25 theorems, 17 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Introduction and Mathematical Formulation
  3. Typical Methods for Solving the MEB Problem
  4. Promise Problems, Property Testing and Partitioning Theorems
  5. Enclosing Theorems and the Computation of the Diameter of a Set
  6. Concluding Remarks

Key Result

Theorem 1

Any point in the convex hull of a finite point set in $\mathbb{R}^d$ is a convex combination of some at most $d + 1$ of these points.

Theorems & Definitions (25)

  • Theorem 1: Carathéodory's theorem (1907) Caratheodory1907
  • Lemma 2: Elzinga-Hearn lemma (1972) ElzingaH1972
  • Lemma 3: Elzinga-Hearn lemma (1972) ElzingaH1972
  • Theorem 4: Elzinga-Hearn theorem (1972) ElzingaH1972
  • Theorem 5: Elzinga-Hearn theorem (1972) ElzingaH1972
  • Proposition 6: Bădoiu-Clarkson proposition (2003) BadoiuC2003
  • Lemma 7: Gonçalves-Keren-Shahar-Yehuda lemma (2023) GoncalvesKSY2023
  • Theorem 8: Helly's partitioning theorem (1913) Helly1923
  • Theorem 9: Tverberg's partitioning theorem (1966) Tverberg1966
  • Theorem 10: Radon's partitioning theorem (1921) Radon1921
  • ...and 15 more