Approximability of convex bodies and volume entropy in Hilbert geometry (Q514631)

The notion of approximability was introduced in [\textit{R. Schneider} and \textit{J. A. Wieacker}, Bull. Lond. Math. Soc. 13, 149--156 (1981; Zbl 0421.52005)] and it measures in some sense how well a convex set can be approximated by polytopes. In this paper, the author gives some connections between the volume entropy and approximability. It is proved that the volume entropy is twice the approximability for a Hilbert geometry in dimension two or three, as well as in higher dimensions the approximability is a lower bound of the entropy. Also, the author shows the existence of Hilbert geometries with intermediate volume growth and the volume entropy is not a limit, in general.