The approximability of a convex body is a number which measures the difficulty in
approximating that convex body by polytopes. In the interior of a convex body one
can define its Hilbert geometry. We prove on the one hand that the volume entropy is
twice the approximability for a Hilbert geometry in dimension two or three, and on
the other hand that in higher dimensions the approximability is a lower bound of the
entropy. As a corollary we solve the volume entropy upper bound conjecture in
dimension three and give a new proof in dimension two different from the one given in
(Pacific J. Math. 245:2 (2010), 201–225). Moreover, our method allows us to
prove the existence of Hilbert geometries with intermediate volume growth on
the one hand, and that in general the volume entropy is not a limit on the
other hand.
