Convex hulls of uniform samples from a convex polygon (Q2898908)

Let \(N_n\) be the number of vertices of the convex hull of a sample of size \(n\), drawn uniformly from the interior of a convex polygon with \(r\) vertices. It was shown in [\textit{P. Groeneboom}, Probab. Theory Relat. Fields 79, No. 3, 327--368 (1988; Zbl 0635.60012)] that \(\{N_n -{{2r}\over{3}} \log n\}/\{{{10r}\over{27}}\log n\}^{1/2}\to {\mathcal N}(0, 1)\) in distribution, where \({\mathcal N}(0, 1)\) denotes the standard normal distribution. The central limit result for \(N_n\) was subsequently derived from a corresponding result for the boundary of the convex hull of the approximating Poisson point process. \textit{A. V. Nagaev} and \textit{I. M. Khamdanov} extended in an unpublished paper [``Limit theorems for functionals of random convex hulls'' (Russian), Preprint. Institute of Mathematics, Academy of Sciences of Uzbekistan] the central limit theorem for the number of verticies \(N_n\) from [P. Groeneboom, loc. cit.]. NEWLINENEWLINEThe aim of this paper to give a simple proof of the result from [A. V. Nagaev and I. M. Khamdanov, loc. cit.].