Abstract
Some properties and applications of generalized convex sets and generalized convex functions in multidimensional real, complex, and hypercomplex spaces are described.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Leichthweis, Convex Sets. Nauka, Moscow, 1985 (in Russian).
Yu.B. Zelins’kyi, Convexity. Selected Chapters. Institute of Mathematics, NAS of Ukraine, Kyiv, 2012 (in Russian).
Yu.B. Zelins’kyi, Multi-Valued Mappings in Analysis. Naukova Dumka, Kyiv, 1993 (in Russian).
H. Behnke and E. Peschl, “Zur Theorie der Funktionen mehrerer komplexer Veränderlichen Konvexität in bezug auf analytische Ebenen im kleinen und großen,” Math. Ann., 111(2), 158–177 (1935).
A. Martineau, “Sur la topologie des espaces de fonctions holomorphes convexité linéele,” Math. Ann., 163(1), 62–88 (1966).
L.A. Aizenberg, “Decomposition of holomorphic functions of several complex variables into partial fractions,” Sib. Math. J., 8, 859–872 (1967).
L.A. Aizenberg, A.P. Yuzhakov, and L.Ya. Makarova, “On linear convexity in ℂn,” Sib. Mat. Zh., 9(4), 731–746 (1968).
Yu.B. Zelins’kyi, I.Yu. Vygovs’ka, and M.V. Stefanchuk, “Shadow problem,” Dopov. NAN Ukrainy, 5, 15–20 (2015).
Yu.B. Zelins’kyi, I.Yu. Vygovs’ka, and M.V. Stefanchuk, “Generalized convex sets and the problem of shadow,” Ukr. Math. J., 67, 1874–1883 (2016).
G. Khudayberganov, On the homogeneously polynomially convex hull of the union of balls. Manuscript dep. at VINITI 02/21/1982 , No. 1772, 85 Dep, 1982 (in Russian).
Y.B. Zelins’kyi and M.V. Stefanchuk, “Generalizations of the shadow problem,” Ukr. Math. J., 68, 862–867 (2016).
M.V. Tkachuk and T.M. Osipchuk, “Shadow problem for an ellipsoid of revolution,” Zbirn. Pratz Inst. Matem. NAN Ukrainy, 12(3), 243–250 (2015).
T.M. Osipchuk, “The shadow problem for domains on a plane,” Pratzi IPMM NAN Ukrainy, 30, 100–105 (2016).
T.M. Osipchuk and M.V. Tkachuk, “The shadow problem for domains in Euclidean spaces,” Ukr. Mat. Visn., 13(4), 532–542 (2016). (in Russian).
T.M. Osipchuk, “Some remarks about systems of balls that create a shadow at a point,” Pratzi IPMM NAN Ukrainy, 31, 109–116 (2017).
Yu.B. Zelins’kyi, I.Yu. Vygovs’ka, and Kh.K. Dakhil, “The shadow problem for balls of fixed radius,” Ukr. Mat. Visn., 13(4), 599–602 (2016).
Yu.B. Zelins’kyi, “The shadow problem for a family of sets,” Zbirn Prats Inst Matem. NAN Ukrainy, 12(4), 197–204 (2015).
Yu.B. Zelins’kyi and H.K. Dakhil, “About one shadow problem for balls of fixed radius,” Pratzi IPMM NAN Ukrainy, 30, 75–81 (2016).
T.M. Osipchuk, “On system of balls with equal radii generating shadow at point,” Bulletin de la société des sciences et des lettres de Lódź, 68(2), 77–84 (2018).
I.L. Kantor and A.S. Solodovnikov, Hypercomplex Numbers. Nauka, Moscow, 1973 (in Russian).
A.S. Besicovitch and R. Rado, “A plane set of measure zero containing circumferences of every radius,” J. London Math. Soc., 43, 717–719 (1968).
A.S. Besicovitch, “On Kakeya’s problem and a similar one,” Math. Zeit, 27, 312–320 (1928).
A.S. Besicovitch, “Sur deux questions d’integrabilite des fonctions,” J. Soc. Phys., Math., Perm, 2, 105–123 (1919).
A.S. Besicovitch, “On fundamental geometric properties of plane line-sets,” J. London Math. Soc., 39, 441–448 (1964).
M.V. Stefanchuk and M.V. Tkachuk, “A null-measured set containing spheres of arbitrary radii,” Zbirn Prats Inst Matem. NAN Ukrainy, 12(4), 285–289 (2015).
G.A. Mkrtchyan, On Hypercomplex Convex Sets. Preprint 87.42. Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, Kyiv, 1987 (in Russian).
G.A. Mkrtchyan, “On strong hypercomplex convexity,” Ukr. Mat. Zhur., 42(2), 182–187 (1990).
Yu.B. Zelins’kyi and G.A. Mkrtchyan, “On extreme points and hypercomplex convex domains,” Dokl. Acad. Nauk SSSR, 311(6), 1299–1302 (1990).
M.V. Stefanchuk, “Extreme elements in hypercomplex space,” Dop. NAN Ukrainy, 4, 13–19 (2016).
M.V. Stefanchuk, “Generalization of the concept of convexity in a hypercomplex space,” Bulletin de la société des sciences et des lettres de Lódź, 68(2), 85–94 (2018).
M.V. Stefanchuk and M.V. Tkachuk, “Linearly convex and conjugate functions in hypercomplex space,”
Zbirn Prats Inst Matem. NAN Ukrainy, 12(3), 225–235 (2015).
32. M.V. Stefanchuk, “Properties of conjugate functions in hypercomplex space,” Pratzi Mizhnarodn. Geometr. Tzentru, 10(2), 39–46 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Professor Yurii Borysovych Zelins’kyi
Presented by O. Dovhoshiy
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 21, No. 2, pp. 255–278, April–June, 2024.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Stefanchuk, M.V. On generalized convex sets and their applications. J Math Sci 284, 383–399 (2024). https://doi.org/10.1007/s10958-024-07357-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07357-w