The farthest point map on the 4-cube

TL;DR

This paper investigates the dynamics of the farthest point map on the 4-cube's boundary, revealing its limit set structure and connections to unfoldings, and explores intrinsic geometric measures like radius and diameter.

Contribution

It characterizes the limit set of the farthest point map on the 4-cube and introduces a loose definition of star unfolding for 4-polytopes, extending previous understanding from lower dimensions.

Findings

Limit set of the 4-cube's farthest point map is union of diagonals of eight facets.

Limit set differs from that of simplices, being more complex.

Intrinsic radius/diameter ratio likely decreases with dimension.

Abstract

We study the farthest point mapping on (the boundary of) the 4-cube with respect to the intrinsic metric, and its dynamics as a multivalued mapping. It is a piecewise rational map. It is more complicated than the one on the 3-cube, but it is shown that the limit set of the farthest point map on the 4-cube is the union of the diagonals of eight (3-cube) facets, like the farthest point map on the 3-cube whose limit set is the union of the six (square) facets. This is in contrast to the doubly covered simplices and (the boundary of) the regular 4-simplex, where the limit set is a finite set. If the source point is in the interior of a facet, its limit set is also in the facet. The farthest point mapping is closely related to the star unfolding and source unfolding. We give a loose definition of star unfolding of the surface of a 4-dimensional polytope. We also study the intrinsic radius and diameter of the 4-cube. It is expected that the intrinsic radius/diameter ratio of an n-cube is monotonically decreasing in dimension.