Nonstandard $n$-distances based on certain geometric constructions

TL;DR

This paper explores generalized $n$-distances derived from geometric constructions, highlighting their computational challenges and connections to graph theory concepts like Steiner and minimal spanning trees.

Contribution

It provides new examples of $n$-distances based on geometry and reveals their links to important graph theoretical measures.

Findings

Computation of the best constant for $n$-distances can be complex.

Euclidean Steiner tree length is an example of an $n$-distance.

Minimal spanning tree length also serves as an $n$-distance.

Abstract

The concept of $n$-distance was recently introduced to generalize the classical definition of distance to functions of $n$ arguments. In this paper we investigate this concept through a number of examples based on certain geometrical constructions. In particular, our study shows to which extent the computation of the best constant associated with an $n$-distance may sometimes be difficult and tricky. It also reveals that two important graph theoretical concepts, namely the total length of the Euclidean Steiner tree and the total length of the minimal spanning tree constructed on $n$ points, are instances of $n$-distances.