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Equichordal Points of Convex Bodies

Leo Jang, Donghan Kim

TL;DR

The paper resolves the classical equichordal point problem in $n$-dimensional convex bodies for $n \ge 2$ by a topological argument. It constructs a map $\varphi: \mathrm{Int}X \to \mathcal{C}(\mathbb{S}^{n-1}, \mathbb{R})$ encoding fixed half-chord lengths and proves $\varphi$ is continuous and injective. A hypothetical pair of equichordal points would yield an antipodal function $g$ whose zero-set is forced by the $\text{Borsuk-Ulam}$ theorem, leading to a contradiction. Consequently, multiple equichordal points cannot exist, establishing a definitive nonexistence result with a clean topological obstruction. The approach highlights the power of antipodal arguments in convex geometry and the utility of function-space representations in geometric problems.

Abstract

The equichordal point problem is a classical question in geometry, asking whether there exist multiple equichordal points within a single convex body. An equichordal point is defined as a point through which all chords of the convex body have the same length. This problem, initially posed by Fujiwara and further investigated by Blaschke, Rothe, and Weitzenböck, has remained an intriguing challenge, particularly in higher dimensions. In this paper, we rigorously prove the nonexistence of multiple equichordal points in $n$-dimensional convex bodies for $n \geq 2$. By utilizing topological tools such as the Borsuk-Ulam theorem and analyzing the properties of continuous functions and mappings on convex bodies, we resolve this long-standing question.

Equichordal Points of Convex Bodies

TL;DR

The paper resolves the classical equichordal point problem in -dimensional convex bodies for by a topological argument. It constructs a map encoding fixed half-chord lengths and proves is continuous and injective. A hypothetical pair of equichordal points would yield an antipodal function whose zero-set is forced by the theorem, leading to a contradiction. Consequently, multiple equichordal points cannot exist, establishing a definitive nonexistence result with a clean topological obstruction. The approach highlights the power of antipodal arguments in convex geometry and the utility of function-space representations in geometric problems.

Abstract

The equichordal point problem is a classical question in geometry, asking whether there exist multiple equichordal points within a single convex body. An equichordal point is defined as a point through which all chords of the convex body have the same length. This problem, initially posed by Fujiwara and further investigated by Blaschke, Rothe, and Weitzenböck, has remained an intriguing challenge, particularly in higher dimensions. In this paper, we rigorously prove the nonexistence of multiple equichordal points in -dimensional convex bodies for . By utilizing topological tools such as the Borsuk-Ulam theorem and analyzing the properties of continuous functions and mappings on convex bodies, we resolve this long-standing question.
Paper Structure (7 sections, 3 theorems, 20 equations)

This paper contains 7 sections, 3 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonexistence of Multiple Equichordal Points in Convex Bodies
  4. Continuity of $\varphi(x)$ for each $x \in \mathrm{Int} X$
  5. Injectivity of $\varphi$
  6. Continuity of $\varphi$
  7. Nonexistence of Multiple Equichordal Points

Key Result

Theorem 2.1

The are no nonconstant antipodal continuous map $f:\mathbb{S}^n \to \mathbb{R}^k$ for every $n,k \in \mathbb{N}$ with $k\leq n$.

Theorems & Definitions (5)

  • Theorem 2.1: Borsuk-Ulam
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5