The known relation between $\alpha$-sets and Vial weakly convex sets in Euclidean spaces of dimension greater than two is strengthened. Namely, in the formula describing the relationship between nonconvexity measures for $\alpha$-sets and weakly convex sets, the double Chebyshev radius is replaced by the diameter of the set with a coefficient. However, in the two-dimensional space the corresponding estimate is expressed in terms of the set diameter without a coefficient and is more accurate. In this connection, the question of the possibility of further refinement of the estimate for the nonconvexity degree $\alpha$ in terms of the weak convexity parameter $R$ and the set diameter in Euclidean spaces of dimension greater than two remains open.
Keywords: $\alpha$-set, weakly convex set, generalized convex set, diameter of a set, Chebyshev radius, convex hull
Received May 30, 2024
Revised July 9, 2024
Accepted July 15, 2024
Funding Agency: This work was supported by the Russian Science Foundation (project no. 24-21-00424, https://rscf.ru/en/project/24-21-00424/).
Vladimir Nikolaevich Ushakov, Corresponding member of RAS, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: ushak@imm.uran.ru
Aleksandr Anatol’evich Ershov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: ale10919@yandex.ru
REFERENCES
1. Uspenskii A.A., Ushakov V.N., Fomin A.N. $\alpha$-Mnozhestva i ikh svoystva [$\alpha$-Sets and their properties]. Yekaterinburg, Deposited at VINITI on 02.04.2004, № 543-V2004, 62 p.
2. Zelinskii Yu.B. Vypuklost’. Izbrannyye glavy [Convexity. Selected topics]. Kyiv, Inst. Math. NASU Publ., 2012, 280 p. doi: 10.13140/RG.2.1.4282.0641
3. Michael E. Paraconvex sets. Math. Scand., 1959, vol. 7, no. 2, pp. 372–376.
4. Ngai H.V., Penot J.-P. Paraconvex functions and paraconvex sets. Studia Math., 2008, vol. 184, no. 1, pp. 1–29.
5. Semenov P.V. Functionally paraconvex sets. Math. Notes, 1993, vol. 54, iss. 6, pp. 1236–1240. doi: 10.1007/BF01209085
6. Ivanov G.E. Slabo vypuklyye mnozhestva i funktsii: teoriya i prilozheniya [Weakly convex sets and functions: theory and applications]. Moscow, Fizmatlit Publ., 2006, 352 p. ISBN: 978-5-9221-0738-9 .
7. Ushakov V.N., Ershov A.A. Estimation of the growth of the degree of nonconvexity of reachable sets in terms of $\alpha$-sets. Dokl. Math., 2020, vol. 102, no. 3, pp. 532–537. doi: 10.1134/S1064562420060198
8. Ushakov V.N., Ershov A.A., Matviychuk A.R. On estimating the degree of nonconvexity of reachable sets of control systems. Proc. Steklov Inst. Math., 2021, vol. 315, suppl. 1, pp. 247–256. doi: 10.1134/S0081543821050199
9. Ushakov V.N., Uspenskii A.A. Theorems on the separability of $\alpha$-sets in Euclidean space. Proc. Steklov Inst. Math., 2017, vol. 299, suppl. 1, pp. 231–245. doi: 10.1134/S0081543817090255
10. Polovinkin E.S., Balashov M.V. Elementy vypuklogo i sil’no vypuklogo analiza [Elements of convex and strongly convex analysis]. Moscow, Fizmatlit, 2007, 440 p. ISBN: 978-5-9221-0896-6 .
11. Ushakov V.N., Uspenskii A.A., Ershov A.A. Alpha-sets in finite-dimensional Euclidean spaces and their applications in control theory. Vestn. St.-Peterbg. Univ. Applied Mathematics. Computer Science. Control Processes, 2018, vol. 14, iss. 3, pp. 261–272. doi: 10.21638/11701/spbu10.2018.307
12. Garkavi A.L. On the Chebyshev center and convex hull of a set. Usp. Mat. Nauk, 1964, vol. 19, no. 6, pp. 139–145.
Cite this article as: V.N. Ushakov, A.A. Ershov. On the relation between $\alpha$-sets and weakly convex sets. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 276–285.
