On the diminishing process of Bálint Tóth (Q2822721)

Let \(K\) and \(K_0\) be convex bodies in \(\mathbb{R}^d\), such that \(K\) contains the origin, and define the process \((K_n, p_n)\), \(n\geq 0\), as follows: let \(p_{n+1}\) be a uniform random point in \(K_n\), and set \(K_{n+1} = K_n\cap(p_{n+1}+K)\). Clearly, \((K_n)\) is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in \(\mathbb{R}^d\).NEWLINENEWLINEIn [\textit{G. Ambrus} et al., Stat. Probab. Lett. 82, No. 1, 191--195 (2012; Zbl 1230.60020)], the process in one dimension when \(K= K_0 = [-1, 1]\) was investigated. In this case the limit object is a random unit interval, whose center has the arcsine distribution (see Theorem 1 in [loc. cit.]). Moreover, in Theorem 2 in [loc. cit.] it is shown that if \(r_n\) is the radius of the interval \(K_n\), then \(4n(r_n - 1/2)\) convergence in distribution to an exponential random variable.NEWLINENEWLINEIn the present paper, the authors analyze the diminishing process in more general case. Section 2 considers the case, when instead of choosing \(p_{n+1}\) uniformly in the interval, the authors choose it according to a translated and scaled version of a fixed distribution \(F\). Again the limit object is a random unit interval. Theorem 2.5 states that for an appropriate choice of \(F\) the distribution of the center has the beta law. In Sections 3 and 4, the authors consider the case when \(K=K_0\) is a cube and a regular \(d\)-dimensional simplex, respectively. The main result of this part is that the center of the limit simplex in barycenter coordinates has multidimensional Dirichlet law, which is a natural generalization of the beta laws to any dimension. The rate of the process is also determined.NEWLINENEWLINEThe processes considered thus far are self-similar in the sense that at each step the process is a scaled and translated version of the original one. Sections 5, 6 and 7 consider diminishing processes in the plane. In the case of the pentagon process, even the shape of the limiting object is random. It is proved that it is a pentagon with equal angles, however it is not regular a.s. This process is not self-similar, and its behavior is more complicated. Finally, in Section 7, the authors consider regular polygons with an odd number of vertices, i.e., \(K=K_0\) is a regular polygon. Using the theory of stochastic orderings for random vectors, they prove that the rate of the speed is \(n^{-1/2}\). The authors conjecture that in the case when the number of vertices is even the speed of the process is \(n^{-1}\). This is established in the case of the square, but in general it is an open problem.