Let X 1 , . . . , X n be independent random points that are distributed according to a probability measure on R d and let P n be the random convex hull generated by X 1 , . . . , X n (n ≥ d + 1). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of P n is strictly monotonically increasing in n.

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein's method from probability theory.

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The random polytope K n , defined as the convex hull of n points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central limit theorems for the intrinsic volumes of K n are presented. A normal approximation bound from Stein's method and estimates for surface bodies are among the involved tools.Intrinsic volumes have been studied extensively in the alternative setting of random polytopes that arise as convex hulls of points chosen uniformly at random inside a fixed convex body. Results concerning the expectation of V ℓ (K n ), ℓ ∈ {1, . . . , d}, have been studied, for example, by Reitzner [10], variance bounds can be found in Böröczky, Fodor, Reitzner and Vígh [4] and Bárány, Fodor and Vígh [1], and central limit theorems were treated in Reitzner [11], Vu [17], Lachièze-Rey, Schulte and Yukich [7] and Thäle, Turchi and Wespi [16]. More details can be found in the references therein.On the other hand, the approximation of a convex body K, by means of a sequence of random polytopes K n , is improved whenever the vertices of K n are restricted to lie on the boundary of K, therefore making it a model worth studying. Indeed, in this framework the expectations ofd}, have been studied, for example, by Buchta, Müller and Tichy [5], Reitzner [8], Schütt and Werner [14] and Böröczky, Fodor and Hug [3]. However, more detailed informations are only known about the distribution of the volume V d (K n ). In particular, an upper variance bound was found by Reitzner [9] and a lower variance bound together with concentration inequalities by Richardson, Vu and Wu [12]. Only recently, Thäle [15] obtained a quantitative central limit theorem for V d (K n ) based on Stein's method.Our first aim is to generalize the results obtained in [9, 12] to V ℓ (K n ), ℓ ∈ {1, . . . , d}. In fact, we prove lower variance bounds following the ideas of [1,11,12] and upper variance bounds in the manner of [1], making use of the Efron-Stein jackknife inequality from [9]. In particular, the upper variance bounds imply strong laws of large numbers as in [1]. Secondly, we prove quantitative central limit theorems for V ℓ (K n ), ℓ ∈ {1, . . . , d}, using a normal approximation bound obtained in [6], extending the result of [15]. We now introduce some notation in order to present our results. Let (a n ) n∈N and (b n ) n∈N be two sequences of real numbers. We write a n ≪ b n (or a n ≫ b n ) if there exist a constant c ∈ (0, ∞) and a positive number n 0 such that a n ≤ c b n (or a n ≥ c b n ) for all n ≥ n 0 . Furthermore, a n = Θ(b n ) means that b n ≪ a n ≪ b n .Our first result concerns asymptotic lower and upper bounds, respectively, for the variances of the intrinsic volumes.Theorem 1.1. Let K ∈ K 2 + and choose n random points on ∂K independently and according to the probability distribution H d−1 . Then,Based on a result stated in [8, Theorem 1] concerning the behaviour of V ℓ (K) − E[V ℓ (K n )], the upper variance bounds of Theorem 1.1 ...

For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d ≥ 3, we establish an exact formula for the average number of (d − 1)-dimensional facets that can be defined by d points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when d = 3, a case which is of particular interest for applications in biophysics, chemistry and polymer science.We also show that the asymptotical average number of facets behaves as, where T is the total duration of the motion and ∆t is the minimum time lapse separating points that define a facet.

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