Equichordal Points of Convex Bodies

TL;DR

This paper proves that in any convex body of dimension two or higher, there cannot be more than one equichordal point, resolving a long-standing problem in geometric theory using topological methods.

Contribution

The paper provides a rigorous proof that multiple equichordal points cannot exist in higher-dimensional convex bodies, settling a classical geometric question.

Findings

Proves the nonexistence of multiple equichordal points in convex bodies for n ≥ 2.

Utilizes topological tools like the Borsuk-Ulam theorem in geometric context.

Resolves a longstanding open problem in convex geometry.

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 Weitzenb\"ock, 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.