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Random inscribed polytopes

Abstract

For convex bodies K with C2 boundary and everywhere positive Gauß-Kronecker curvature in Rd, we explore random polytopes Kn with the n vertices chosen uniformly along the boundary of K. In particular, we determine the asymptotic properties of the volume of these random polytopes when n is large. We provide results concerning the variance and higher moments of this functional. Previously, these results are considered very difficult to obtain due to the high technicalities in the existing integral methods. We will demonstrate here a different method for obtaining such estimates, namely the so-called divide-and-conquer martingale technique. We first give a concentration result for Vold(Kn) which indicates the behavior of exponential decay of the deviation of volume from its mean. This result not only implies the upper bound on the variance of Vold(Kn) previously obtained by Reitzner [54] via refinement of integral methods, it also gives us an upper bound on any k-th moments of the volume for k 2 expressed in terms of the variance. Then we give a matching lower bound on the variance, which is tight up to a multiplicative constant factor that depends only on the fixed dimension d and the convex body K. Lastly, we show that central limit theorem holds asymptotically for the volume functional of our inscribing model provided that the random polytopes are constructed with vertices chosen on the boundary of K according to the Poisson Process

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