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Towards the mathematical foundation of the minimum enclosing ball and related problems

Michael N. Vrahatis

TL;DR

This work surveys and consolidates the mathematical foundations of the minimum enclosing ball problem in Euclidean spaces, situating MEB among a broad family of enclosing, width, and clustering problems. It foregrounds key geometric quantities—circumradius ${\rho_{cir}}$, inradius ${\rho_{inr}}$, diameter, and width—and collects classical bounds (notably Jung's and Steinhagen's theorems) while deriving explicit radii for regular simplices. The paper also links MEB to core convexity results via Carathéodory, Helly, Radon, and Tverberg, including modern no-dimension extensions, and discusses inner/outer radii of compact convex bodies and regular polytopes (Brandenberg–Klee, Perelman, González Merino), highlighting both sharp bounds and computational complexity. By compiling these enclosing/partitioning theorems, the work provides a rigorous framework for analyzing MEB and related problems across spaces and geometries, with implications for clustering, optimization, and geometric computation. The intended impact is to support precise, theory-backed algorithmic approaches and to guide extensions to non-Euclidean settings and large-scale data contexts, including core-sets and property-testing perspectives.

Abstract

Theoretical background is provided towards the mathematical foundation 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. The study of several problems that are similar or related to the minimum enclosing ball problem has received a considerable impetus from the large amount of applications of these problems in various fields of science and technology. The proposed theoretical framework is based on several enclosing (covering) and partitioning (clustering) theorems and provides among others bounds and relations between the circumradius, inradius, diameter and width of a set. These enclosing and partitioning theorems are considered as cornerstones in the field that strongly influencing developments and generalizations to other spaces and non-Euclidean geometries.

Towards the mathematical foundation of the minimum enclosing ball and related problems

TL;DR

This work surveys and consolidates the mathematical foundations of the minimum enclosing ball problem in Euclidean spaces, situating MEB among a broad family of enclosing, width, and clustering problems. It foregrounds key geometric quantities—circumradius , inradius , diameter, and width—and collects classical bounds (notably Jung's and Steinhagen's theorems) while deriving explicit radii for regular simplices. The paper also links MEB to core convexity results via Carathéodory, Helly, Radon, and Tverberg, including modern no-dimension extensions, and discusses inner/outer radii of compact convex bodies and regular polytopes (Brandenberg–Klee, Perelman, González Merino), highlighting both sharp bounds and computational complexity. By compiling these enclosing/partitioning theorems, the work provides a rigorous framework for analyzing MEB and related problems across spaces and geometries, with implications for clustering, optimization, and geometric computation. The intended impact is to support precise, theory-backed algorithmic approaches and to guide extensions to non-Euclidean settings and large-scale data contexts, including core-sets and property-testing perspectives.

Abstract

Theoretical background is provided towards the mathematical foundation 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. The study of several problems that are similar or related to the minimum enclosing ball problem has received a considerable impetus from the large amount of applications of these problems in various fields of science and technology. The proposed theoretical framework is based on several enclosing (covering) and partitioning (clustering) theorems and provides among others bounds and relations between the circumradius, inradius, diameter and width of a set. These enclosing and partitioning theorems are considered as cornerstones in the field that strongly influencing developments and generalizations to other spaces and non-Euclidean geometries.
Paper Structure (12 sections, 28 theorems, 35 equations)

This paper contains 12 sections, 28 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Preliminary Background Material
  4. Uniqueness, Related Problems, Theory and Applications of the MEB Problem
  5. Upper Bounds of Radii of Convex Sets and Compact Convex Bodies
  6. Relations and Upper Bounds of Circumradius, Inradius, Diameter and Width of Convex Sets
  7. Explicit Upper Bounds of Radii, Width and Median of Regular Simplices
  8. Inner and Outer Radii of Compact Convex Bodies and Regular Polytopes
  9. Mathematical Approaches for Enclosing and Partitioning a Set
  10. Theorems of Jung, Steinhagen and Perel${}^\prime\!$man
  11. Theorems of Carathéodory, Helly, Radon and Tverberg
  12. Concluding Remarks

Key Result

Theorem 1

Let $\tau^d$ be a regular $d$-simplex in ${\mathbb{R}}^d$ with diameter ${\rm diam}(\tau^d)$, then the circumradius $\rho_{\rm cir}^d (\tau^d)$ of $\sigma^d$ coincides with the barycentric circumradius $\beta_{\rm cir}^d (\tau^d)$ and

Theorems & Definitions (28)

  • Theorem 1: Circumradius of regular simplices Vrahatis2023
  • Theorem 2: Inradius of regular simplices Vrahatis2023
  • Theorem 3: Width of regular simplices Vrahatis2023Har-PeledR2023
  • Theorem 4: Medians of regular simplices Vrahatis2023
  • Theorem 5: Generalization of Pythagoras' theorem Vrahatis2023
  • Theorem 6: Perel${}^\prime\!$man-Pukhov theorem Perelman1987
  • Theorem 7: Jung's enclosing theorem (1901) Jung1901
  • Theorem 8: Steinhagen's theorem (1921) Steinhagen1921
  • Theorem 9: Perel${}^\prime\!$man's enclosing theorem (1987) Perelman1987
  • Theorem 10: Blumenthal-Wahlin enclosing theorem (1941) BlumenthalW1941
  • ...and 18 more