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

Flag-approximability of convex bodies and volume growth of Hilbert geometries