In this paper we generalize some of the classical results of Sulanke (1963), (1964) in the context of spindle convexity. A planar convex disc S is spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.
Let K be a d dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the s-th intrinsic volumes Vs(Kn) of Kn for s ∈ {1, . . . , d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the Economic Cap Covering Theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.
NotationWe shall work in d-dimensional Euclidean space R d , with origin o, and scalar product ·, · , and induced norm · . The dimension d will be fixed throughout the paper. We shall not distinguish between the Euclidean space and the underlying vector space, and we will use the words point and vector interchangeably, as we need them. Points of R d are denoted by small-case letters of the roman alphabet, and sets by capitals. For reals we use either Greek letters or small-case letters. B j stands for the j-dimensional ball of radius 1 centered at the origin, S j−1 denotes the boundary of B j and κ j denotes the volume of B j . Note that any point x ∈ ∂B j = S j−1 can be considered as a point of the boundary of B j and also as an outer normal to B j at the point x. For a point set T ⊂ R d , we denote the convex hull of T by conv T or simply by [T ]. A compact convex set K with nonempty interior is called a convex body.The intrinsic volumes V s (K), s = 0, . . . , d of a convex body K can be introduced as coefficients of the Steiner formulawhere K + λB d is the Minkowski sum of K and λB d of radius λ ≥ 0. In particular, V d is the volume functional, V 0 (K) = 1, V 1 is proportional to the *
a b s t r a c tLet K be a convex body in R d and let X n = (x 1 , . . . , x n ) be a random sample of n independent points in K chosen according to the uniform distribution. The convex hull K n of X n is a random polytope in K , and we consider its mean width W (K n ). In this article, we assume that K has a rolling ball of radius > 0. First, we extend the asymptotic formula for the expectation of W (K ) − W (K n ) which was earlier known only in the case when ∂K has positive Gaussian curvature. In addition, we determine the order of magnitude of the variance of W (K n ), and prove the strong law of large numbers for W (K n ). We note that the strong law of large numbers for any quermassintegral of K was only known earlier for the case when ∂K has positive Gaussian curvature.
Let Ξ 0 = [−1, 1], and define the segments Ξ n recursively in the following manner: for every n = 0, 1, . . . , let Ξ n+1 = Ξ n ∩ [a n+1 − 1, a n+1 + 1], where the point a n+1 is chosen randomly on the segment Ξ n with uniform distribution. For the radius ρ n of Ξ n we prove that n(ρ n − 1/2) converges in distribution to an exponential law, and we show that the centre of the limiting unit interval has arcsine distribution.
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