Geodesic
Joint distribution of the number of vertices and the area of convex hulls generated by a uniform distribution in a convex polygon
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 2, pp. 230-241

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A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process.
Keywords: convex hull, central limit theorem.
Mots-clés : convex polygon, Poisson point process, binomial point process 
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     author = {Isakjan M. Khamdamov and Zoya S. Chay},
     title = {Joint distribution of the number of vertices and the area of convex hulls generated by a uniform distribution in a convex polygon},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {230--241},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2021_14_2_a10/}
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Isakjan M. Khamdamov; Zoya S. Chay. Joint distribution of the number of vertices and the area of convex hulls generated by a uniform distribution in a convex polygon. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 2, pp. 230-241. http://geodesic.mathdoc.fr/item/JSFU_2021_14_2_a10/