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Some results about equichordal convex bodies

Jesús Jerónimo-Castro, Francisco G. Jimenez-Lopez, Efrén Morales-Amaya

TL;DR

The paper studies equichordal convex bodies, where an interior body $L$ is equichordal for $K$ if every chord of $K$ tangent to $L$ has a fixed length $λ$. Using isoptic curves, rotor geometry, and Fourier analysis of the support function, it shows that in the plane there exist infinitely many non-circular $L$ with the equichordal property, while in dimensions $n\ge3$ the only possibility is concentric Euclidean balls. It further develops connections between isoptics and rotors in polygons, and proves higher-dimensional rigidity results: if $K$ admits an equichordal $L$ in $\mathbb{R}^n$, then under natural sectional conditions $K$ and $L$ are concentric balls. Additional 3D isoptic-sphere rigidity statements reinforce the spherical symmetry phenomenon. Overall, the work clarifies when tangent-chord length constraints enforce spherical symmetry and illuminates links between constant width, isoptics, and rotor configurations.

Abstract

Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 2$, with $L\subset \text{int}\, K$. We say that $L$ is an equichordal body for $K$ if every chord of $K$ tangent to $L$ has length equal to a given fixed value $λ$. J. Barker and D. Larman proved that if $L$ is a ball, then $K$ is a ball concentric with $L$. In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. We also prove some results about isoptic curves and give relations between isoptic curves and convex rotors in the plane.

Some results about equichordal convex bodies

TL;DR

The paper studies equichordal convex bodies, where an interior body is equichordal for if every chord of tangent to has a fixed length . Using isoptic curves, rotor geometry, and Fourier analysis of the support function, it shows that in the plane there exist infinitely many non-circular with the equichordal property, while in dimensions the only possibility is concentric Euclidean balls. It further develops connections between isoptics and rotors in polygons, and proves higher-dimensional rigidity results: if admits an equichordal in , then under natural sectional conditions and are concentric balls. Additional 3D isoptic-sphere rigidity statements reinforce the spherical symmetry phenomenon. Overall, the work clarifies when tangent-chord length constraints enforce spherical symmetry and illuminates links between constant width, isoptics, and rotor configurations.

Abstract

Let and be two convex bodies in , , with . We say that is an equichordal body for if every chord of tangent to has length equal to a given fixed value . J. Barker and D. Larman proved that if is a ball, then is a ball concentric with . In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. We also prove some results about isoptic curves and give relations between isoptic curves and convex rotors in the plane.

Paper Structure

This paper contains 6 sections, 11 theorems, 41 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminary concepts
  3. Some results about isoptic curves in the plane
  4. Some comments about rotors in the plane
  5. An inequality about the length of some special chords
  6. Some results in higher dimensions

Key Result

Theorem 1

Let $K$ be a strictly convex body in the plane with differentiable boundary and let $\alpha\in(0,\pi)$ be a fixed angle such that $\frac{\alpha}{\pi}$ is an irrational number. Suppose $c(t)=c_0$, for every $t\in [0,2\pi]$, for a positive number $c_0.$ Then $K$ is a disc.

Figures (12)

  • Figure 1: Parameters of the isoptic curve
  • Figure 2: A convex body with two isoptics with corresponding values of $c(t)$ both of constant value for the angles $\alpha_1=\frac{\pi}{5}$ and $\alpha_2=\frac{3\pi}{5}$
  • Figure 3: The angle between the chord $[\gamma_{\alpha}(t+\pi-\alpha),\gamma_{\alpha}(t)]$ and the tangent lines are equal
  • Figure 4: The value of $h(t)$ is constant
  • Figure 5: The value of $d(t)$ is constant
  • ...and 7 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Theorem 6
  • Theorem 7
  • Corollary 1
  • ...and 2 more