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Shortest closed curve to contain a sphere in its convex hull

Mohammad Ghomi, James Wenk

Abstract

We show that in Euclidean 3-space any closed curve $γ$ which contains the unit sphere within its convex hull has length $L\geq4π$, and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in $n$ dimensions, we include the estimate $L\geq Cn\sqrt{n}$ by Nazarov, which is sharp up to the constant $C$.

Shortest closed curve to contain a sphere in its convex hull

Abstract

We show that in Euclidean 3-space any closed curve which contains the unit sphere within its convex hull has length , and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in dimensions, we include the estimate by Nazarov, which is sharp up to the constant .
Paper Structure (5 sections, 12 theorems, 20 equations, 2 figures)

This paper contains 5 sections, 12 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Minimal Inspection Curves
  3. Unfolding
  4. Spiral Decomposition
  5. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $\gamma\colon [a,b]\to\mathbf{R}^3$ be a closed rectifiable curve of length $L$, and $r$ be the inradius of the convex hull of $\gamma$. Then Equality holds only if, up to a reparameterization, $\gamma$ is simple, $\mathcal{C}^{1,1}$, lies on a sphere of radius $\sqrt2 \,r$, and traces consecutively $4$ semicircles of length $\pi r$.

Figures (2)

  • Figure 1: The baseball curve
  • Figure 2: Construction of the competing curve

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 11 more