Random polytopes in smooth convex bodies

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Let K ⊂ Rd be a convex body and choose points xl, x2, …, xn randomly, independently, and uniformly from K. Then Kn = conv {x1, …, xn} is a random polytope that approximates K (as n → ∞) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K – vol Kn when K is a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).