  The approximability of a convex body is a number which measures the
difficulty to approximate that body by polytopes. We prove that twice the
approximability is equal to the volume entropy for a Hilbert geometry in
dimension two end three and that in higher dimension it is a lower bound of the
entropy.
  As a corollary we solve the entropy upper bound conjecture in dimension three
and give a new proof in dimension two from the one found in
Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).
