Geodesic
On properties of intersection of $\alpha$-sets
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 55 (2020), pp. 79-92

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In this paper, we study the properties of $\alpha$-sets, which are one of the generalizations of convex sets. In the first part of the paper, the equivalence of two definitions of $\alpha$-sets in the plane is proved. The second part of the work is devoted to the experimental study of the properties of simply connected intersections of $\alpha$-sets. It follows from the results of numerical experiments that the value $\alpha$ of the measure of nonconvexity in a simply connected intersection of two $\alpha$-sets can be greater than the initial value of $\alpha$ in intersected sets even when these values are very close to zero. Based on these results, we can hypothesize that, firstly, such an increase in the value of $\alpha$ is possible with an arbitrarily small initial $\alpha$ for intersected sets, secondly, this increase is limited by a linear function of the initial value of $\alpha$.
Keywords: generalized convex set, intersection of sets.
Mots-clés : $\alpha$-set 
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     author = {A. A. Ershov and O. A. Kuvshinov},
     title = {On properties of intersection of $\alpha$-sets},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {79--92},
     publisher = {mathdoc},
     volume = {55},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2020_55_a5/}
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A. A. Ershov; O. A. Kuvshinov. On properties of intersection of $\alpha$-sets. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 55 (2020), pp. 79-92. http://geodesic.mathdoc.fr/item/IIMI_2020_55_a5/