We consider the problem of shadow in the Lobachevsky space. This problem can be treated as the problem of finding conditions guaranteeing that points belong to the generalized convex hull of a family of sets. We determine the limit values of the parameters for which the same configurations of balls guarantee that a point belongs to the generalized convex hull of balls in the Euclidean and hyperbolic spaces. Parallel with families of balls, we consider families of horoballs, as well as certain combinations of balls and horoballs.
Similar content being viewed by others
References
G. Khudaiberganov,On the Homogeneous Polynomially Convex Hull of a Union of Balls[in Russian], Deposited at VINITI, 21, 1772–1785 (1982).
Yu. B. Zelinskii, I. Yu. Vygovskaya, and H. K. Dakhil, “Problem of shadow and related problems,” Proc. Internat. Geom. Center, 9, No. 3-4, 50–58 (2016).
Yu. B. Zelinskii, I. Yu. Vygovskaya, and M. V. Stefanchuk, “Generalized convex sets and the problem of shadow,” Ukr. Mat. Zh., 67, No. 12, 1658–1666 (2015); English translation: Ukr. Math. J., 67, No. 12, 1874–1883 (2016).
Yu. B. Zelins’kyi and M. V. Stefanchuk, “Generalization of the shadow problem,” Ukr. Mat. Zh., 68, No. 6, 757–762 (2016); English translation: Ukr. Math. J., 68, No. 6, 862–867 (2016).
Y. B. Zelinskii, “Generalized convex envelopes of sets and the problem of shadow,” J. Math. Sci., 211, No. 5, 710–717 (2015).
Y. B. Zelinskii, “Problem of shadow (complex case),” Adv. Math.: Sci. J., 5, No. 1, 1–5 (2016).
Y. B. Zelinskii, “The problem of the shadows,” Bull. Soc. Sci. Lett. L´od´z, S´er. Rech. D´eform., 66, No. 1, 37–42 (2016).
Yu. B. Zelinskii, “Shadow problem for a family of sets,” in: Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine, 12, No. 4, 197–204 (2015).
Yu. B. Zelinskii, I. Yu. Vygovskaya, and M. V. Stefanchuk, “Problem of shadow,” Dop. Nats. Akad. Nauk Ukr., No. 5, 15–19 (2015).
T. M. Osipchuk and M. V. Tkachuk, “The problem of shadow for domains in Euclidean spaces,” Ukr. Mat. Visn., 13, No. 4, 532–542 (2016). English translation: J. Math. Sci., 224, No. 4, 555–562 (2017)).
M. V. Tkachuk and T. M. Osipchuk, “Problem of shadow for an ellipsoid of revolution,” in: Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine, 12, No. 3 (2015), pp. 243–250.
Zh. Kaidasov and E. V. Shikin, “On the isometric immersion of a convex domain of the Lobachevsky plane containing two horodisks in E3,” Mat. Zametki, 39, No. 4, 612–617 (1986).
B. A. Rozenfel’d, Non-Euclidean Spaces [in Russian], Nauka, Moscow (1977).
N. M. Nestorovich, Geometric Structures in the Lobachevskii Plane [in Russian], Gostekhteorizdat, Moscow (1951).
A.V. Kostin, “Problem of shadow in the Lobachevskii space,” Ukr. Mat. Zh., 70, No. 11, 1525–1532 (2018); English translation: Ukr. Math. J., 70, No. 11, 1758–1766 (2018).
A.V. Kostin and I. K. Sabitov, “Smarandache theorem in hyperbolic geometry,” J. Math. Phys., Anal., Geom., 10, No. 2, 221–232 (2014).
A.V. Kostin, “On the asymptotic lines on pseudospherical surfaces,” Vladikavkaz. Mat. Zh., 21, No. 1, 16–26 (2019).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 1, pp. 61–68, January, 2021. Ukrainian DOI: 10.37863/umzh.v73i1.2397.
Rights and permissions
About this article
Cite this article
Kostin, A. Some Generalizations of the Shadow Problem in the Lobachevsky Space. Ukr Math J 73, 67–75 (2021). https://doi.org/10.1007/s11253-021-01908-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-021-01908-z