Farthest points on flat surfaces

TL;DR

This paper investigates the set of points farthest from a given point on flat surfaces, analyzing their properties and structure within the context of geometric distance functions.

Contribution

It characterizes the structure of farthest points on flat surfaces, providing new insights into their geometric configuration and properties.

Findings

Identifies the set of farthest points for arbitrary points on flat surfaces.

Provides a detailed description of the geometric structure of these farthest points.

Enhances understanding of distance functions on flat surfaces.

Abstract

We consider the distance function from an arbitrary point $p$ on a flat surface, and determine the set $F_{p}$ of all \emph{farthest points} (i.e., points at maximal distance) from $p$.