Expensive Homeomorphism of Convex Bodies

TL;DR

The paper resolves whether an expansive homeomorphism can exist on any $n$-dimensional convex body $C\subset \mathbb{R}^n$ by proving nonexistence for all $n$. It combines topological tools, including the Borsuk–Ulam and Brouwer fixed-point theorems, with a constructive boundary-to-ball lifting: any boundary homeomorphism on $\mathbb{S}^n$ fixing a point $e$ extends to a homeomorphism $F:B^n\to B^n$ that commutes with the canonical quotient. An inductive dimension argument then shows that no expansive $f: B^n\to B^n$ can exist, since such an $f$ would yield an expansive boundary map extendable to $F$, contradicting the induction base. The result provides a complete resolution to Klee's question and clarifies the constraints imposed by convexity on expansive dynamics in compact convex sets, highlighting a deep link between convex geometry and topological dynamics.

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.

Theorems & Definitions (8)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3: Brouwer fixed-point theorem
• Theorem 2.4: Borsuk-Ulam
• Theorem 3.1
• proof
• Theorem 3.2
• proof