On the area and perimeter of a random convex hull in a bounded convex set

Abstract

Suppose u is a compact convex set in R 2 and i Y 1 i n, is a random sample of points in the interior of u. Under general assumptions on u and the distribution of the i we study the asymptotic properties of certain statistics of the convex hull of the sample.

Limit theorems for certain functionals of unions of random closed sets

“…In the particular two-dimensional case very strong asymptotic theorems have been obtained during the past decades since Groeneboom's work [4] followed by a number of other authors (see e.g. [1], [2], [5] and [7]). In higher dimensions the results obtained so far are much weaker, nevertheless important progress has been made in in vestigating the asymptotics of the volume outside the convex hull (see [8] and the references therein).…”

“…It requires an additional regularity assumption (REG) which says, roughly speaking, that if closed sets F u F 2 E & are such that the probabilities T x (Fi) = P{X П Fi ф 0}, % = 1,2, are small enough then the probability P{X П Fi Ф 0 Л J П F 2 ф 0} becomes negligibly small. To formulate the condition denote for F 1 …”

Length and surface area estimation under smoothness restrictions

“…The perimeter and surface area of the convex hull of the sample can Length and surface area estimation SGSA • 349 be successfully used for estimating the length and surface area of G. See [2] and [9] for more details. However, assuming that G is convex may be too restrictive in practice.…”

The Normalized Graph Cut and Cheeger Constant: From Discrete to Continuous

“…the number of connected components [5], the intrinsic dimensionality [29] and, more generally, the homology [10], [11], [13], [14], [34], [37], [44]; the Minkowski content [17], as well as the perimeter and area (volume) [8]. In the The normalized graph cut and Cheeger constant SGSA • 911 related field of stochastic geometry, Khmaladze and Weil [27] established limit properties of Poisson point processes in the context of the change-set problem, where the Poisson process has two homogeneous components, with different intensities inside and outside of an unknown convex compact subset of R d .…”