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Approximation of spherical convex bodies of constant width $π/2$

Huhe Han

TL;DR

The paper addresses approximating spherical convex bodies of constant width $\frac{\pi}{2}$ on $\mathbb{S}^2$ by spherical polytopes preserving width, in the Hausdorff metric. It develops a constructive boundary-modification approach that preserves constant width via supporting hemispheres and spherical polarity, producing a sequence of width-$\frac{\pi}{2}$ approximants and controlling their proximity to the original body. The main contribution is proving the existence of spherical polytopes $\mathcal{P}_{\varepsilon}$ with $h(C, \mathcal{P}_{\varepsilon})\leq \varepsilon$ for any $\varepsilon>0$, thereby solving the $\tau=\frac{\pi}{2}$ case in dimension $n=2$. This extends Blaschke-type approximation results to the critical spherical width, enabling discrete, width-preserving representations of spherical bodies with potential applications in spherical geometry and related computational tasks.

Abstract

Let $C\subset \mathbb{S}^2$ be a spherical convex body of constant width $τ$. It is known that (i) if $τ<π/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $τ$ whose boundary consists only of arcs of circles of radius $τ$ such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$; (ii) if $τ>π/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $τ$ whose boundary consists only of arcs of circles of radius $τ-\fracπ{2}$ and great circle arcs such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$. In this paper, we present an approximation of the remaining case $τ=π/2$, that is, if $τ=π/2$ then for any $\varepsilon>0$ there exists a spherical polytope $\mathcal{P}_\varepsilon$ of constant width $π/2$ such that the Hausdorff distance between $C$ and $\mathcal{P}_\varepsilon$ is at most $\varepsilon$.

Approximation of spherical convex bodies of constant width $π/2$

TL;DR

The paper addresses approximating spherical convex bodies of constant width on by spherical polytopes preserving width, in the Hausdorff metric. It develops a constructive boundary-modification approach that preserves constant width via supporting hemispheres and spherical polarity, producing a sequence of width- approximants and controlling their proximity to the original body. The main contribution is proving the existence of spherical polytopes with for any , thereby solving the case in dimension . This extends Blaschke-type approximation results to the critical spherical width, enabling discrete, width-preserving representations of spherical bodies with potential applications in spherical geometry and related computational tasks.

Abstract

Let be a spherical convex body of constant width . It is known that (i) if then for any there exists a spherical convex body of constant width whose boundary consists only of arcs of circles of radius such that the Hausdorff distance between and is at most ; (ii) if then for any there exists a spherical convex body of constant width whose boundary consists only of arcs of circles of radius and great circle arcs such that the Hausdorff distance between and is at most . In this paper, we present an approximation of the remaining case , that is, if then for any there exists a spherical polytope of constant width such that the Hausdorff distance between and is at most .
Paper Structure (3 sections, 8 theorems, 16 equations, 2 figures)

This paper contains 3 sections, 8 theorems, 16 equations, 2 figures.

Table of Contents

  1. introduction
  2. preliminaries
  3. Proof of Theorem \ref{['theorem3']}

Key Result

Theorem 1

For any spherical convex body $C\subset \mathbb{S}^2$ of constant width $\tau<\pi/2$, and for any $\varepsilon>0$ there exists a body $C_\varepsilon$ of constant width $\tau$ whose boundary consists only of arcs of circles of radius $\tau$ such that where $h(C_1, C_2)$ means the Hausdorff distance between $C_1$ and $C_2$ .

Figures (2)

  • Figure 1: A spherical convex body $C_\varepsilon^1.$
  • Figure 2: $\Delta(H(R)\cap H(R_1))<\varepsilon$.

Theorems & Definitions (10)

  • Theorem 1: Lassak22-1
  • Theorem 2: hanam
  • Theorem 3
  • Proposition 1: nishimurasakemi2
  • Lemma 2.1: hanam
  • Proposition 2: hwam
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof