  Suppose we choose $N$ points uniformly randomly from a convex body in $d$
dimensions. How large must $N$ be, asymptotically with respect to $d$, so that
the convex hull of the points is nearly as large as the convex body itself? It
was shown by Dyer-F\"uredi-McDiarmid that exponentially many samples suffice
when the convex body is the hypercube, and by Pivovarov that the Euclidean ball
demands roughly $d^{d/2}$ samples. We show that when the convex body is the
simplex, exponentially many samples suffice; this then implies the same result
for any convex simplicial polytope with at most exponentially many faces.
