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. $\endgroup$ – Nate Eldredge 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.


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