On properties of intersection of $\alpha$-sets

A. A. Ershov ; O. A. Kuvshinov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 55 (2020), pp. 79-92.

Résumé

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 $\alpha$.

Détail

Keywords: generalized convex set, intersection of sets.
Mots-clés : $\alpha$-set

Bibliographie

[1] Uspenskii A. A., Ushakov V. N., Fomin A. N., $\alpha$-sets and their properties, Deposited in VINITI 02.04.2004, No 543-B2004, IMM UB RAS, Ekaterinburg, 2004, 62 pp. (in Russian)

[2] Ivanov G. E., Weak convex sets and functions: theory and applications, Fizmatlit, M., 2006

[3] Zelinskii Yu. B., Convexity. Selected chapters, Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, 2012

[4] Michael E., “Paraconvex sets”, Mathematica Scandinavica, 7:2 (1959), 312–315 | DOI | MR

[5] Ngai H. V., Penot J.-P., “Paraconvex functions and paraconvex sets”, Studia Mathematica, 184:1 (2008), 1–29 | DOI | MR | Zbl

[6] Semenov P. V., “Functionally paraconvex sets”, Mathematical Notes, 54:6 (1993), 74–81 | DOI | MR | Zbl

[7] Ershov A. A., Pershakov M. V., “On matching up the alpha-sets with other generalizations of convex sets”, VI Information school of a young scientist, Central Scientific Library of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 2018, 143–150 (in Russian) | DOI

[8] Ivanov G. E., “Weakly convex sets and their properties”, Mathematical Notes, 79:1–2 (2006), 55–78 | DOI | DOI | MR | Zbl

[9] Ivanov G. E., Polovinkin E. S., “Second-order convergence of an algorithm calculating the value of linear differential games”, Doklady Mathematics, 51:1 (1995), 29–32 | MR | MR | Zbl

[10] Ivanov G. E., Golubev M. O., “Strong and weak convexity in nonlinear differential games”, IFAC PapersOnline, 51:32 (2018), 13–18 | DOI

[11] Polovinkin E. S., Balashov M. V., Elements of convex and nonconvex analisys, Fizmatlit, M., 2007

[12] Ershov A. A., Ushakov V. N., “An approach problem for a control system with an unknown parameter”, Sbornik: Mathematics, 208:9 (2017), 1312–1352 | DOI | MR | Zbl

[13] Ushakov V. N., Uspenskii A. A., Ershov A. A., “Alpha sets in finite-dimensional euclidean spaces and their applications on control theory”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 14:3 (2018), 261–272 (in Russian) | DOI | MR

[14] Kuvshinov O. A., A software package for calculating the intersection of two polygons approximating two circles with cut out sectors, Certificate of state registration of a computer program. Reg. no. AAAA-G19-619112690043-6 dated November 26, 2019

[15] Ershov A. A., A software package for calculating alpha nonconvexity measures for alpha sets, Certificate of state registration of a computer program. Reg. No AAAA-G19-619112690042-9 dated November 26, 2019

[16] Ushakov V. N., Uspenskii A. A., “$\alpha$-sets in finite Euclidean spaces and theor properties”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 26:1 (2016), 95–120 (in Russian) | DOI | MR | Zbl