The surface area of a polyhedron in a normed space is defined as the sum of the areas of its faces, each divided by the area of the central section of the unit ball, parallel to the face. This functional naturally extends to convex bodies. In this paper, it is proved, in particular, that the surface area of the unit sphere in any three-dimensional normed space does not exceed 8. Bibliography: 2 titles.
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I. K. Babenko, “Asymptotic volume of tori and geometry of convex bodies,” Mat. Zametki, 44, 177–190 (1988).
V. V. Makeev, “On parallelepipeds and centrally symmetric hexagonal prisms circumscribed around a three-dimensional centrally symmetric convex body,” Zap. Nauchn. Semin. POMI, 372, 103–107 (2009).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 415, 2013, pp. 21–23.
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Makeev, V.V., Nikanorova, M.Y. Estimating the Surface Area of Spheres in Normed Spaces. J Math Sci 212, 531–532 (2016). https://doi.org/10.1007/s10958-016-2681-9
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DOI: https://doi.org/10.1007/s10958-016-2681-9