Abstract
We overview the main ideas and techniques of the functional-analytical approach to some extremal problems of convex geometry that stem from the Queen Dido problem.
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References
Manin Y.I., A Course of Mathematical Logic for Mathematicians, New York: Springer, 2010.
Kelvin B., “Isoperimetric problems,” In: Kelvin B. and Thompson Nature series, vol. II, Ch. Isoperimetrical Problems, pp. 571-592, Macmillan and Co., London, 1891
Kutateladze S. S. and Rubinov A. M.“The Minkowski duality and its applications,” Russian Math. Surveys, 27:3, 137–191 (1972).
Reshetnyak Yu. G. On the Length and Swerve of a Curve and the Area of a Surface, (Ph. D. Thesis, unpublished), Leningrad State University, Leningrad, 1954.
Loomis L. “Unique direct integral decomposition on convex sets,” Amer. Math. J., 84:3, 509–526 (1962).
Cartier P., Fell J. M., and Meyer P. A.“Comparaison des mesures poertées par un ensemble convexe compact,” Bull. Soc. Math. France, 64, 435–445 (1964).
Kutateladze S. S. “Positive Minkowski-linear functionals over convex surfaces,” Soviet Math. Dokl. 11:3, 767–769 (1970).
Kutateladze S. S. “Choquet boundaries in K-spaces,” Russian Math. Surveys, 30:4, 115–155 (1975).
Alexandrov A. D. Selected Works. Part 1: Selected Scientific Papers, Gordon and Breach, London etc. (1996).,
Firey W. “Blaschke sums of convex bodies and mixed bodies,” in: it Proc. of the Colloquium on Convexity, 1965,l Kobenhavns Univ. Mat. Inst., Copenhagen, (1967), 94–101.
Schneider R. Convex Bodies: The Brunn–Minkowski Theory, Cambridge University Press, Cambridge (1993).
Gardner R. J. “The Brunn–Minkowski Inequality,” Bull. Amer. Math. Soc., 39:3, 355–405 (2002).
Urysohn P. S. “Interdependence between the integral breadth and volume of a convex body,” Mat. Sb., 31:3, 477–486 (1924).
Rosales C. “Isoperimetric regions in rotationally symmetric convex bodies,” Indiana Univ. Math. J., 51:5, 1201–1214 (2003).
Ritore M. and Rosales C. “Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones,” Trans. Amer. Math. Soc., 356:11, 4601–4622 (2004).
Zălinesku C. Convex Analysis in General Vector Spaces, World Scientific, New Jersey, etc. (2002).
Göpfert A., Tammer Ch., Riahi H., and Zălinesku C. Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York (2003).
Jahn J. Vector Optimization: Theory, Applications, and Extensions,Berlin, Springer-Verlag (2004).
Kusraev A. G. and Kutateladze S. S. Subdifferential Calculus: Theory and Applications, Nauka, Moscow (2007).
Isenberg C. The Science of Soap Films and Soap Bubbles, Dover Publications, New York (1992).
Lovett D. Demonstrating Science with Soap Films, IOP Publishing Ltd., Bristol and Philadelphia (1994).
Pogorelov A. V. “Imbedding a ‘soap bubble’ into a tetrahedron,” Math. Notes, 56:2, 824–826 (1994).
Kutateladze S. S. “Multiobjective problems of convex geometry,” Sib. Math. J., 50:5, 887–897 (2010).
Kutateladze S. S. “Multiple criteria problems over Minkowski balls,” J. Appl. Ind. Math., 7:2, 208–214 (2013).
Funding
The research was supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement 075–02–2021–1844) and carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF–2022–0004).
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To Anatoly Kusraev with great respect.
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Kutateladze, S.S. DIDO’S PROBLEM AND BEYOND. J Math Sci 271, 778–785 (2023). https://doi.org/10.1007/s10958-023-06899-9
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DOI: https://doi.org/10.1007/s10958-023-06899-9