  We introduce the flag-approximability of a convex body to measure how easy it
is to approximate by polytopes. We show that the flag-approximability is
exactly half the volume entropy of the Hilbert geometry on the body, and that
both quantities are maximized when the convex body is a Euclidean ball.
  We also compute explicitly the asymptotic volume of a convex polytope, which
allows us to prove that simplices have the least asymptotic volume.
