Some Results on the Minkowski Difference in Topological Vector Spaces
DOI:
https://doi.org/10.56143/ujmcs.v2i2.4/Keywords:
topological vector space, Minkowski difference, support functions , class of compact convex sets, continuous mappingAbstract
This work highlights important topological properties of Minkowski operations defined in topological vector spaces. The distributivity of Minkowski sum and difference operations with respect to the Cartesian product of sets is proved. Conditions under which the density property of sets is preserved under Minkowski operations are obtained. Furthermore, it is established as a theorem that the Minkowski difference operation is continuous in the class of compact convex sets, and this is proved using topological methods.
References
[1] Bogachev, V.I., Smolyanov, O.G.: Topological vector spaces and their applications. Springer, Cham (2017).
[2] Narici, L., Beckenstein, E.: Topological vector spaces. 2nd ed. CRC Press, Boca Raton (2024).
[3] Rockafeller, T.: Convex analysis. Princeton University Press, New Jersey (1997).
[4] Schneider, R.: Convex bodies: the Brunn–Minkowski theory. Cambridge University Press, Cambridge (2014).
[5] Dvurechensky, P.E., Ivanov, G.E.: Algorithms for computing Minkowski operators and their application in differential games. Comput. Math. Math. Phys. 54 (2), 235–264 (2014). https://doi.org/10.1134/S0965542514020055
[6] Ivanov, G.E.: Weakly convex sets and their properties. Math. Notes 79 (1–2), 55–78 (2006). https://doi.org/10.1007/s11006-006-0005-y
[7] Nuritdinov, J.T.: On the Minkowski difference of triangles. Bull. Inst. Math. 4 (6), 50–57 (2021).
[8] Mamatov, M.S., Nuritdinov, J.T., Turakulov, K.S., Mamazhonov, S.M.: Geometric properties of the Minkowski operator. Bull. Karaganda Univ. Math. Ser. 11 (4), 127–137 (2024).
[9] Nuritdinov, J., Kakharov, S., Tashxodjayev, A.: Application of Minkowski operator in artificial intelligence tasks. AIP Conf. Proc. 3244 (1), 030016 (2024).
[10] Courteaux, M., Ramlot, B., Lambert, P., Van Wallendael, G.: Lightweight implicit approximation of the Minkowski sum of an N-dimensional ellipsoid and hyperrectangle. Mathematics 13 (8), 1326 (2025). https://doi.org/10.3390/math13081326
[11] Polovinkin, E.S., Balashov, M.V.: Elements of convex and strongly convex analysis. Fizmatlit, Moscow (2007).
