### 50 years sets with positive reach -- a survey.

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Negative association for a family of random variables $\left({X}_{i}\right)$ means that for any coordinatewise increasing functions f,g we have $({X}_{i\u2081},...,{X}_{{i}_{k}})g({X}_{j\u2081},...,{X}_{{j}_{l}})\le f({X}_{i\u2081},...,{X}_{{i}_{k}})g({X}_{j\u2081},...,{X}_{{j}_{l}})$ for any disjoint sets of indices (iₘ), (jₙ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem Saxena and Joag-Dev Proschan, and brought to convex geometry in 2005 by Wojtaszczyk Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of...

We show that whenever the $q$-dimensional Minkowski content of a subset $A\subset {\mathbb{R}}^{d}$ exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ${\mathbb{R}}^{d}$, $d\ge 3$.

Jensen et al. (1990) gave an exact expression for the κ-function in non-overlapping Boolean models. The present study proposes and evaluates an approximate expression for the κ-function in overlapping isotropic Boolean models based on an approximation of the covariogram of the primary grain. We study the suitability of a Boolean model for two binary images using this approximate expression.

Bertrand's paradox is a longstanding problem within the classical interpretation of probability theory. The solutions 1/2, 1/3, and 1/4 were proposed using three different approaches to model the problem. In this article, an extended problem, of which Bertrand's paradox is a special case, is proposed and solved. For the special case, it is shown that the corresponding solution is 1/3. Moreover, the reasons of inconsistency are discussed and a proper modeling approach is determined by careful examination...

This paper presents distributional properties of a random cell structure which results from a growth process. It starts at the points of a Poisson point process. The growth is spherical with identical speed for all points; it stops whenever the boundaries of different cells have contact. The whole process finally stops after time t. So the space is not completely filled with cells, and the cells have both planar and spherical boundaries. Expressions are given for contact distribution functions,...

Problems related to the random approximation of convex bodies fall into the field of integral geometry and geometric probabilities. The aim of this paper is to give a survey of known results about the stochastic model that has received special attention in the literature and that can be described as follows:Let K be a d-dimensional convex body in Eucliden space Rd, d ≥ 2. Denote by Hn the convex hull of n independent random points X1, ..., Xn distributed identically and uniformly in the interior...