Abstract In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector through asymmetric metrics polyhedral norms by means nonstandard methods, that is, ultrapowers the at hand. The carry structure stratified with strata indexed dual faces unit ball. Explicit neighborhood bases descriptions horofunctions are provided.

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path from s to t in a polyhedral terrain. We give some properties of such paths. In the case where the face sequence is specified, we show that the shortest descending path is unique, and giv...