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Universal Chord Theorem and a Topological Analysis

Ion Ciudin, Eugen J. Ionascu

TL;DR

This paper advances the understanding of the Universal Chord Theorem by characterizing which chords can be isolated and by analyzing the structure of the open additive sets $H(f)^*$. Building on Hopf-type arguments, it shows that points in ${\cal P}\setminus\{1\}$ cannot be isolated, while carefully constructed chords can realize isolated points for other lengths, via explicit Hopf-set constructions. It develops a general framework for realizing $H(f)^*$ from open additive sets $U$, and provides concrete results in the cases $m=1$ and $m=2$, including a symmetry about $x=\tfrac{1}{2}$ and a formula linking $H(f)$ to $V$ when $H(f)^*=V\cup(1,\infty)$. The work uncovers rich connections to number theory, topology, and descriptive set theory, and proposes open questions and conjectures (e.g., Conjecture K) to guide further exploration of maximal Hopf sets and their integer-structure properties.

Abstract

We study the set of chords of a real-valued continuous function on [0,1] with f(0)=f(1)=0. We describe which chords may appear as isolated points and provide examples illustrating our characterization. Maximal Hopf sets are introduced and analyzed.

Universal Chord Theorem and a Topological Analysis

TL;DR

This paper advances the understanding of the Universal Chord Theorem by characterizing which chords can be isolated and by analyzing the structure of the open additive sets . Building on Hopf-type arguments, it shows that points in cannot be isolated, while carefully constructed chords can realize isolated points for other lengths, via explicit Hopf-set constructions. It develops a general framework for realizing from open additive sets , and provides concrete results in the cases and , including a symmetry about and a formula linking to when . The work uncovers rich connections to number theory, topology, and descriptive set theory, and proposes open questions and conjectures (e.g., Conjecture K) to guide further exploration of maximal Hopf sets and their integer-structure properties.

Abstract

We study the set of chords of a real-valued continuous function on [0,1] with f(0)=f(1)=0. We describe which chords may appear as isolated points and provide examples illustrating our characterization. Maximal Hopf sets are introduced and analyzed.
Paper Structure (6 sections, 11 theorems, 29 equations)

This paper contains 6 sections, 11 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Points in ${\cal P}\setminus \{1\}$ cannot be isolated
  3. Possible ways to construct of $H(f)^*$
  4. Case $\bf m=1$.
  5. Case $\bf m=2$.
  6. Conclusions and open questions

Key Result

Theorem 1

For a given length $\ell$, the necessary and sufficient condition for a continuous function $f:[0,L]\to \mathbb R$ with $f(0)=f(L)$ to have a horizontal chord of length $\ell$ is that $\ell=\frac{L}{m}$ for some positive integer $m$.

Theorems & Definitions (18)

  • Theorem 1: Universal Cord Theorem
  • Proposition 1: Burns
  • Theorem 2: Hopf
  • Theorem 3: Diana2024
  • Definition 1
  • Proposition 2
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 8 more