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 midpoints of all line segments having one endpoint in $B_1$ and another endpoint in $B_2$, is never greater than $\nu$?


**Motivation.**
Given two subsets $A$ and $B$ of a  geodesic metric space $(X,d)$, let  $(A + B)/2$ denote the set of midpoints of all minimal geodesic segments with
one endpoint in $A$ and another endpoint in $B$. 

Given a Borel measure $\mu$ on $X$, let us say that the metric-measure space $(X,d,\mu)$ is  a *Brunn-Minkowski space* if whenever $A$ and $B$ are open subsets of $X$ with finite measure, then
$$
\mu \left( \frac{A + B}{2}\right) \geq \sqrt{\mu(A)\mu(B)} .
$$


Loosely speaking, this is the sort of inequality that after the work of [Cordero-Erausquin, McCann and Schmuckenschläger][1], [Sturm][2], and [Lott and Villani][3] (and many others) has come to be associated with *non-negative Ricci curvature*. In fact, it is the weakest possible related notion I could come up with. *The condition in the OP aims to check for "non-positive" Ricci curvature.*

Hilbert geometries generalize hyperbolic geometry (although they could also be flat, as when $K$ is a simplex), but it is very hard to come up with a good, geometric way to say "they are negatively curved". In many cases they are indeed Gromov hyperbolic (Yves Benoist wrote a fantastically pretty paper on this), but it is also known that if a Hilbert geometry is non-positively curved in the sense of Busemann, then it is hyperbolic space. I'm just trying to come up with something that seems reasonable making use that Hilbert geometries + Hausdorff measure (or Holmes-Thompson volume) are nice metric-measure spaces.

**Remark.** Finsler spaces can be very different from Riemannian spaces. For example, compact, convex surfaces in normed spaces are not necessarily Brunn-Minkowski spaces (and their Ricci curvature in the Sturm-Lott-Villani sense is not non-negative).  C. Vernicos and I found a cute little example of this phenomenon.


  [1]: http://link.springer.com/article/10.1007/s002220100160
  [2]: http://link.springer.com/article/10.1007/s11511-006-0003-7
  [3]: http://annals.math.princeton.edu/2009/169-3/p04