Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Metric Spaces where Geodesics are Never Unique

Amlan Banaji

Abstract

This article concerns a class of metric spaces, which we call multigeodesic spaces, where between any two distinct points there exist multiple distinct minimising geodesics. We provide a simple characterisation of multigeodesic normed spaces and deduce that $(C([0,1]),||\cdot||_1)$ is an example of such a space, but that finite-dimensional normed spaces are not. We also investigate what additional features are possible in arbitrary metric spaces which are multigeodesic.

Metric Spaces where Geodesics are Never Unique

Abstract

This article concerns a class of metric spaces, which we call multigeodesic spaces, where between any two distinct points there exist multiple distinct minimising geodesics. We provide a simple characterisation of multigeodesic normed spaces and deduce that is an example of such a space, but that finite-dimensional normed spaces are not. We also investigate what additional features are possible in arbitrary metric spaces which are multigeodesic.
Paper Structure (3 sections, 11 theorems, 15 equations, 3 figures)

This paper contains 3 sections, 11 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Normed vector spaces
  3. Metric spaces

Key Result

Lemma 3

Let $(X,d)$ be a metric space, let $u,v \in X$, and suppose $\gamma \colon [0,1] \to X$ satisfies $\gamma(0) = u$, $\gamma(1) = v$. Then $\gamma$ is a geodesic if and only if for all $s,t \in [0,1]$ we have $d(\gamma(s),\gamma(t)) \leq |s-t|d(u,v)$.

Figures (3)

  • Figure 1: The space $\mathbb{R}^2$ equipped with the taxicab metric (which is induced by the $1$-norm $||(x,y)||_1 = |x| + |y|$, whose unit ball is shaded grey). This space is geodesic, but neither uniquely geodesic nor multigeodesic. The dashed lines show several different geodesics between $(0,0)$ and $(1,1)$. The dotted line shows the unique geodesic between $(0,0)$ and $(-1,0)$.
  • Figure 2: The Laakso space is multigeodesic. The metric here is different from the one induced by the Euclidean distance; indeed, it is shown in b:lang that the space cannot embed into Euclidean space via a bi-Lipschitz mapping.
  • Figure 3: Some functions lying on two disjoint geodesics between 0 and a function $g$ with $||g||_1 = 1$, in the space $(C[0,1], ||\cdot||_1)$ from Corollary \ref{['cor:cts']}. Left: the geodesic goes through the function $h(x) = xg(x)$. Right: the geodesic goes through the alternative 'intermediate' function $Cg$, where $C = ||h||_1 \approx 0.45$ is such that the two shaded regions have equal areas.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 14 more