Research Article

Random polytopes in smooth convex bodies

Imre Báránya1

a1 The Mathematical Institute of the Hungarian Academy of Sciences, 1364 Budapest P.O.B. 127, Hungary.


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).

(Received August 01 1991)

Key Words:

  • 52A22: CONVEX AND DISCRETE GEOMETRY; General Convexity; Random convex sets and integral geometry