Recall that a *Hilbert geometry* is the interior of a convex body $K \subset \mathbb{R}^n$ provided with the metric
$$
d(x,y) = \frac{1}{2} \ln\left(\frac{|x-b|}{|y-b|}\frac{|y-a|}{|x-a|}\right) ,
$$
where $a$ and $b$ are the points of intersection of the boundary of $K$ and the straight line determined by $x$ and $y$ with the provision that $x$ lie in the segment $ay$ (and $y$ lie in the segment $xb$).

Question.Given two metric balls $B_1$ and $B_2$ of finite Hausdorff measure $\nu > 0$ in a Hilbert geometry $(K,d_K)$, is it true that the Hausdorff measure of $(B_1 + B_2)/2$, defined as the set formed by the midpointsfor the Hilbert metricof all line segments having one endpoint in $B_1$ and another endpoint in $B_2$, is never greater than $\nu$?

The idea is to see if Hilbert geometries behave as Hadamard manifolds in some sense.