An Inscribing Model for Random Polytopes

Abstract

For convex bodies K with C 2 boundary in R d , we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.

Random inscribed polytopes in projective geometries

“…Finally, we will make extensive use of the fact that K σ (n) contains the σ -surface body with overwhelming probability. More precisely, we require the following result from [57,Lemma 4.2].…”

“…We are now ready to adapt to our situation the main lemma [57,Lemma 3.1], that has to be changed in the proof of [57, Theorem 1.1].…”

“…. , n, according to the economic cap covering, see [57,Lemma 6.6], and in each cap C K (x j , h n ) define the sets D i (x j ), i = 0, . .…”

“…In this article we focus on the convex hull of independent and identically distributed points taken from the boundary of a fixed convex body K . This model of a so-called random inscribed polytope in K was investigated in [13,16,51,52,56,57,60,64], mainly from an asymptotic point of view (as the number of points tends to infinity). In particular, in [63] a central limit theorem is proven for the volume of the random inscribed polytope inside a sufficiently smooth convex body in Euclidean space.…”

Random inscribed polytopes in projective geometries

“…Proof. To prove [57, Lemma 3.1] one first shows [57,Claim 8.1], where the first and second moment of the volume are asymptotically bounded. Following the proof of [57, Claim 8.1] and [57,Claim 8.2] we obtain…”