Symmetric matrices and Hilbert's fourth problem

Question

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result:

Theorem. All straight lines are extremals of the variational problem $$ \gamma \mapsto \int_\gamma L \ , $$ where the Lagrangian function $L : T\mathbb{R}^n \rightarrow \mathbb{R}$ is smooth outside the zero section and absolutely homogeneous of degree one (i.e., $L(x,\lambda v) = |\lambda|L(x,v)$), if and only if there exists a smooth function $f: \mathbb{R} \times S^{n-1} \rightarrow \mathbb{R}$ for which $$ L(x,v) = \int_{\xi \in S^{n-1}} |\xi\cdot v|f(\xi\cdot x,\xi)dA(\xi) . $$

In trying to embellish this preprint for submission, I started thinking in how to give nice, (even more) explicit formulas of Finsler metrics and Lagrangians on the tangent space of the $n$-sphere or projective $n$-space for which all great circles or projective lines are extremals. I came up with this construction:

Given a positive definite $n\times n$ symmetric matrix $A$ and a non-zero tangent vector $(x,v)$ to the $(n-1)$-sphere in Euclidean $n$-space, let $\sigma(A;x,v)$ denote the absolute value of the sine of the angle formed by the vectors $Ax$ and $Av$.

Proposition. If $\mu$ is a smooth (signed) measure of compact support on the space $PS(n)$ of symmetric positive-definite $n\times n$ matrices, then all great circles are extremals of the Lagrangian $$ L(x,v) := \int_{A \in PS(n)} \frac{\|Av\|}{\|Ax\|} \sigma(A;x,v) \ d\mu(A) $$ defined on the tangent space of the $(n-1)$-sphere $x_1^2 + \cdots + x_n^2 = 1$.

Proof. A simple computation will show that the integrand is nothing but the pullback of the arc-length element for the standard metric on the sphere under the map $x \mapsto Ax/\|Ax\|$. Geodesics for this Lagrangian are great circles. The set of all $1$-homogeneous Lagrangians on a manifold sharing the same extremals is a linear space and so adding or integrating such Lagrangians yields another of the same kind. Q.E.D.

In other words, the whole point is that in the sphere there are lots of collineations that are not isometries and this can be exploited.

Finally:

Question 1. What are the natural conditions on the measure $\mu$ for which the construction works? I gave "smooth" and "compact support" as conditions, but these are not necessary: use a delta function and the construction gives a nice Riemannian metric.

Question 2. What conditions of the measure will guarantee that $L$ is a Finsler metric (i.e. that $L(x,\cdot)$ be a norm on each tangent space $T_x S^{n-1}$)? Requiring $\mu$ to be non-negative and non-zero will do the trick, but this maybe too strong for $n > 3$.

In fact, this construction hits the same snag as the Busemann-Pogorelov construction: if one uses positive measures, one ends up with hypermetric metrics on the sphere (because the Minkowski sum of ellipsoids---the little something hidden behind the construction---is a zonoid). This brings me to

Question 3. Is every hypermetric Finsler solution of Hilbert's fourth problem on the sphere obtained by the above construction by choosing an appropriate measure $\mu$.

Answer 1

This is a short answer to Question 3 above:

Theorem. Every continuous symmetric distance function on the $n$-sphere for which great circles are geodesic can be obtained as a limit of linear combinations of distance functions of the form $$ d_A(x,y) = \arccos\left(\frac{Ax \cdot Ay}{||Ax|| ||Ay||} \right) \ , $$ where $A$ is an invertible linear map from $\mathbb{R}^{n+1}$ to itself.

Sketch of the proof. For every invertible $A$, the metric $d_A$ is projective (i.e., all great circles are geodesics) and it has an associated Crofton-style formula where the measure on the totally geodesic hyperspheres is simply $A^{-1}_{\#} \mu_0$, where $\mu_0$ is the standard measure on the (dual) $n$-sphere that appears in the Crofton formula for the standard metric on $S^n$ and, abusing notation, $A$ is also used to denote the map that maps geodesic hyperspheres to geodesic hyperspheres induced from the linear map $A$.

Because of the Busemann-Pogorelov solution to Hilbert's fourth all that we need to check is that any signed Borel measure on the sphere that is invariant under the antipodal map can be obtained as a limit of linear combinations of measures of the form $A^{-1}_{\#} \mu_0$ ($A$ invertible linear transformation). This is so because we can obtain the measures $\delta_x + \delta_{-x}$ $(x \in S^n)$ and linear combinations of these measures are dense in the set of of Borel measures invariant under the antipodal map.

This is rather sketchy, but I'll put a link to a preprint where this is all done carefully in a "few" days.