# Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements $\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ is not the arc-length element of a Riemannian metric on the plane). A simple application of this idea is the construction of a class of (Finsler) metrics in projective space for which geodesics are projective lines:

Consider the group $G$ of projective transformations on projective $n$-space and let $\mu$ be a compactly-supported Borel probability measure on $G$. Define the length of a curve $\gamma$ to be the expected value of the (canonical) length of the curve $T(\gamma)$, where $T \in G$.

Claim. The preceding construction defines a length structure on $\mathbb{R}P^n$ for which projective lines are geodesics.

This is essentially the same construction as in this MO question.

We can consider the following abstract version of this construction:

Consider a length space $(X,d)$ and a continuous group action $G \times X \rightarrow X$ of a locally compact topological group $G$ on the space $X$. If $\mu$ is a compactly-supported Borel probability measure on $G$, define the $\mu$-length of a curve $\gamma$ to be the expected value of the (canonical) length of the curve $T(\gamma)$, where $T \in G$, and define the $\mu$-distance between two points $x, y \in X$ as the expected value of $T \mapsto d(Tx,Ty)$.

Questions.

1. Is the length of a curve in $(X,d_\mu)$ equal to its $\mu$-length?

2. When is $(X,d_\mu)$ a length space?

3. If $(X,d)$ is a Riemannian torus with distance function $d$, $G$ is also the torus acting on itself by translations, and $\mu$ is the normalized Haar measure on $G$, is it true that $d_\mu$ the flat metric induced by the stable norm?

Let me sketch why I think this last question may have a positive answer: a characterization of the stable norm by D. Burago states that if $d$ is a periodic length metric (the lift to $\mathbb{R}^n$ of a length metric on $\mathbb{R}^n/\mathbb{Z}^n$), the stable norm is the unique norm satisfying $$\|x-y\| - C \leq d(x,y) \leq \|x-y\| + C$$ for some constant $C > 0$ independent of $x$ and $y$. I hadn't noticed this before, but the set of periodic metrics sharing the preceding inequality is a convex set. Therefore, if an average of periodic length metrics with the same stable norm $\|\cdot\|$ is again a length metric, then the stable norm this average metric will again be $\|\cdot\|$. Averaging the metrics $d(x+z,y+z)$ as $z$ ranges over all points in the unit square should then yield a translation-invariant metric whose stable norm is $\|\cdot\|$. With some luck, this translation-invariant metric is induced by a norm and by Burago's theorem it would have to be the stable norm.

In any case, the above construction allows us to associate a translation invariant metric to a periodic length metric,. The whole question is whether this metric comes from a norm or not.

Disclaimer. I still really haven't thought all these things through, but I thought it would be helpful to write them down to make things clearer (at least for myself) and perhaps someone here has already considered these averages. They are quite natural.

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Just a comment on averaging: sums of intrinsic metrics are usually not intrinsic metrics (they are when, for example, they share the same geodesics). –  alvarezpaiva Jul 16 '13 at 10:02
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