Sub-Finsler horofunction boundaries of the Heisenberg group

TL;DR

This paper characterizes the horofunction boundary for polygonal sub-Finsler metrics on the Heisenberg group, linking it to Pansu derivatives and extending to homogeneous groups.

Contribution

It provides a complete analytic and geometric description of horofunction boundaries for sub-Finsler metrics on the Heisenberg group, connecting to Pansu derivatives.

Findings

Explicit description of horofunction boundary for polygonal sub-Finsler metrics

Connection between horofunctions and Pansu derivatives

Extension of theory to general homogeneous groups

Abstract

We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics---that is, those that arise as asymptotic cones of word metrics---on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.