Abstract
To each convex compact A in Euclidian space En there corresponds a point S (A) from En such that 1) S(x) = x for x ∈ En, 2) S(A + B) = S(A) + S(B), 3) S (Ai) →θ, if Ai converges in the Hausdorff metric to the unit sphere in En, then S(A) is the Steiner point of the set A. The theorem improves certain earlier results on characterizations of the Steiner point.
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Literature cited
B. Grünbaum, Studies in Combinatorial Geometry and the Theory of Convex Bodies [Russian translation] (1971).
R. Schneider, “On Steiner points of convex bodies,” Israel J. Math.,9, pp. 241–249 (1970).
G. T. Sallee, “A noncontinuous ‘Steiner point’,” Israel J. Math.,10, pp. 1–5 (1971).
T. Bonnesen and W. Fenchel, Theorie der Konvexen Körper, Springer-Verlag, Berlin (1943).
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Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 243–247, August, 1973.
In conclusion, I wish to express my appreciation to E. M. Semenov for his constant help with this work.
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Fositsel'skii, E.D. Characterizations of Steiner points. Mathematical Notes of the Academy of Sciences of the USSR 14, 698–700 (1973). https://doi.org/10.1007/BF01147117
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DOI: https://doi.org/10.1007/BF01147117