On weakly m-convex sets

1Dakhil, HK, 2Zelinskii, Yu.B, 2Klishchuk, BA
1Taras Shevchenko National University of Kyiv
2Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2017, 4:3-6
Section: Mathematics
Language: Ukrainian
Some properties of weakly m-convex sets are studied. Various variants of the problem of shadow are investigated. The lower bound for the number of balls that are necessary to create a shadow at a point tangential to sphere $S^{2}$ in the Euclidean space $\mathbb{R}^{3}$, is obtained.
Keywords: 1-hull of a family of sets, conjugate set, Grassmann manifold, m-convex set, problem of shadow, weakly m-convex set
  1. Zelinskii, Yu. B. (2012). Convexity. Selected topics. Kiev: Institute of Mathematics of the NAS of Ukraine (in Russian).
  2. Zelinskii, Yu. B., Momot, I. V. (2001). On the (n, m) convex sets. Ukr. Math. J., 53, No 3, rpp. 422-427 (in Russian). https://doi.org/10.1023/A:1012352607365
  3. Zelinskii, Yu. B., Gretsky, A. S., Momot, I. V. Some results on generalized convex sets. Classical analysis, Proceedings of 10-th intern. sympos. Poland, 1999, pp. 113-124.
  4. Rohlin, V. A., Fuks, D. B. (1977). Basic Topology Course. Moscow: Nauka (in Russian).
  5. Zelinskii, Yu. B. (2015). The problem of shadow for family of sets. Zbirnyk prats' In-tu matematyky NAN Ukrainy, 12, No 4: 197-204 (in Russian).