Expensive Homeomorphism of Convex Bodies

TL;DR

This paper proves that expansive homeomorphisms cannot exist within convex bodies in Euclidean spaces by applying topological theorems and inductive reasoning, revealing fundamental constraints imposed by convexity.

Contribution

It establishes the nonexistence of expansive homeomorphisms in convex bodies using topological tools, providing new insights into the interplay between topology and convex geometry.

Findings

Expansive homeomorphisms do not exist in convex bodies.

Topological theorems like Borsuk-Ulam are key to the proof.

The result applies across all dimensions of Euclidean space.

Abstract

In this paper, we address the longstanding question of whether expansive homeomorphisms can exist within convex bodies in Euclidean spaces. Utilizing fundamental tools from topology, including the Borsuk-Ulam theorem and Brouwer's fixed-point theorem, we establish the nonexistence of such mappings. Through an inductive approach based on dimension and the extension of boundary homeomorphisms, we demonstrate that expansive homeomorphisms are incompatible with the compact and convex structure of these bodies. This work highlights the interplay between topological principles and metric geometry, offering new insights into the constraints imposed by convexity.