From Wikepedia, the definition of geodesic is stated as:

A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that for any $t\in I$ there is a neighborhood $J$ of $t$ in $I$ such that for any $t_1, t_2 \in J$ we have $d(\gamma(t_1), \gamma(t_2)) = v|t_1- t_2|$.

It is a known fact that a metric space, in general, may have no geodesics, except constant curves.

I wonder whether there is a well-defined geodesic in the space of compactly supported functions. If there is, how can we compute it? Is the geodesic unique?

I'm specifically interested in the cases of:

  1. $L^p([0, 1])$ with $L^p$-norm;
  2. $\mathcal{P}([0, 1])$ with sup-norm, where $\mathcal{P}([0, 1])$ denotes the space of piecewise constant functions over $[0, 1]$.

I'm a newbie to the topic of geometry. So any direct answer or reference to this question would be extremely helpful to me!

  • 3
    $\begingroup$ These are vector spaces, so the geodesic is just a straight line, is it not? That is, the geodesic from $f_1$ to $f_2$ is simply $\gamma(t) = (1-t)f_1 + t f_2$. You can verify this from properties of the norm. $\endgroup$ Sep 3 '20 at 17:40
  • $\begingroup$ @NateEldridge: for homogenous metrics that is a geodesic, but it is not necessarily unique. $\endgroup$
    – Ville Salo
    Sep 3 '20 at 17:55
  • $\begingroup$ @NateEldredge That's a good catch. I was too involved thinking about the `advanced' geometric ideas... Is there any way to justify the uniqueness? $\endgroup$
    – mw19930312
    Sep 3 '20 at 18:09
  • $\begingroup$ @mw19930312 I think for $L^p$ it's true and for sup false. At least $l^p$ on $\mathbb{R}^d$ is (Google it), and for sup there's a counterexample on $\mathbb{R}^2$ and you can modify that a bit. $\endgroup$
    – Ville Salo
    Sep 3 '20 at 18:21
  • $\begingroup$ @VilleSalo Thanks for the comments! I'll take a look. $\endgroup$
    – mw19930312
    Sep 3 '20 at 18:25

A good reference for geodesic spaces (which includes a chapter dedicated to normed vector spaces) is Athanase Papadopoulos, Metric Spaces, Convexity, and Nonpositive Curvature (google books has a preview).

As Nate Eldredge pointed out in the comments, in a normed space you can always form a geodesic between two points, say $f$ and $g$, as the affine line $\gamma(t) = (1-t)f + tg$. However, this geodesic is not always unique. In particular, the $L^p$ spaces are uniquely geodesic when $1<p<\infty$, but not for $p=1$ or $p=\infty$. In fact, any function space with the sup-norm is not uniquely geodesic, as the sup-norm is not strictly convex.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.