# Besicovitch's covering theorem for ellipsoids and shadows

The usual Besicovitch's covering theorem concerns closed balls in $\mathbb{R}^d$. It relies on a property called "directionally limited metric space": the principal ingredient is to say that there can't be too many euclidian balls in $\mathbb{R}^d$ with non empty intersection, and such that their respective centers belong to only one ball. The "too many" here is a number that depends only on the dimension $d$.

I am interested in the problem where inside $\mathbb{R}^d$ you consider ellipsoids instead of balls. The ellipsoids I am interested in have a priori unbounded eccentricity, but I am asking a condition of the length of the axis. They have to be of the form $R^{\alpha}$ with, say, $R<1$ and $1<\alpha<\alpha_0$ for some uniform $\alpha_0$.

My question: Would that be possible to play the "directionally limited game" with those hypothesis?

My motivation: I am interested in extending the theory of Borel's differentiation of Radon measure on the sphere at infinity $N(\infty)$ of a simply connected Riemannian manifold $N$ whose sectional curvature is pinched between two negative constants (in my case, in order to simplify, I ask $N$ to cover a closed manifold $M$). This sphere is not a manifold: it only possesses a priori a Hölder structure.

Shadows would play the role of round balls in $\mathbb{R}^d$. To define them, choose $\epsilon>0$, and two points $z\in N$, and $\xi\in N(\infty)$. The shadow $\mathcal{O}_{\epsilon}(\xi,z)$ is defined as the set of extremities of those geodesics starting at $\xi$ and passing through the Riemannian ball centered at $z$ and of radius $\epsilon$.

It occurs that these shadows form a basis of the topology of $N(\infty)$, that they satisfy a covering theorem "à la Vitali" (this is due to Roblin, lemme 1.2.1), and that the Patterson-Sullivan measure satisfies a doubling condition (this is due to Sullivan's shadow lemma). Hence, the theory of Lebesgue's differentiation with respect to the PS measure holds.

For my purpose, I need a covering theorem "à la Besicovitch". And hence I have to play the "directionally limited game" with shadows, where the center of the shadow is played by the geodesic pasing through $\xi$ and $z$. One idea would be, given a family of intersecting shadows whose center only belongs to one shadow, to intersect the corresponding family of cones with a horosphere centered at $\xi$: the intersections would look exaclty like ellipsoids with a "polynomial" eccentricity, with a uniform exponent only depending on the pinching of the curvature. Pulling back by the exponential would enable us to come down to the euclidian case.