Expansive homeomorphisms on convex bodies in dimensions greater than two

Determine whether an n-dimensional convex body C ⊂ R^n with n > 2 admits an expansive homeomorphism T: C → C, meaning a homeomorphism for which there exists a constant ε0 > 0 such that for every distinct pair x ≠ y in C, there exists an integer m with ||T^m(x) − T^m(y)|| ≥ ε0.

Background

The paper investigates whether expansive homeomorphisms can exist on convex bodies, which are compact, convex subsets of Euclidean space with non-empty interior. Expansive homeomorphisms are dynamical systems where points eventually separate by at least a fixed distance under the action of the homeomorphism or its inverse.

The authors note that in one-dimensional cases (intervals), expansive homeomorphisms do not exist, but they state that the existence question remains unresolved in higher dimensions. This open problem is attributed to Klee's list of unsolved problems in geometry, motivating the paper presented in the paper.