Some problems of minima concerning the oval

Tadahiko KUBOTA; Denzaburo HEMMI

doi:10.2969/jmsj/00530372

Tadahiko KUBOTA, Denzaburo HEMMI

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1953 Volume 5 Issue 3-4 Pages 372-389

DOI https://doi.org/10.2969/jmsj/00530372

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Published: 1953 Received: - Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: August 29, 2006 Reason for correction: - Correction: AUTHOR Details: Wrong : Tadahiko KUBOTA, Denzaburo HEMAH
Right : Tadahiko KUBOTA, Denzaburo HEMMI

Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Right : (1) J. Pál: Ein Minimumproblem für Ovale. Math. Ann. 83 (1921). Cf. throughout this note as a reference the excellent report by T. Bonnesen u. W. Fenchel, Theorie der konvexen Körper. (Ergebn, d. Math. III 1.) Berlin (1934).
(2) T. Kubota: Einige Ungleichheitsbeziehungen über Eilinien und Eiflächen. Sci. Rep. Tôhoku Univ. 12 (1923).
(3) M. Yamanouchi: Notes on closed convex figures. Proc. Phys. -Math. Soc. Japan 14 (1932).
(4) Lebesgue: Sur le probléme des isopérimètres et sur les domaines de largeur constante. Bull. Soc. Math. France C. R. (1914).
(5) H. Lebesgue: Sur quelques questions de minimum, relatives aux courbes orbiformes, et sur leurs rapports avec le calcul des variations. J. Math. Pures Appl. 4 (1921).
(6) W. Blaschke: Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts. Math. Ann. 76 (1915).
(7) T. Kubota: Eine Ungleichheit für Eilinien. Math. Z. 20 (1924).
(8) In 1917 (?) Mr. K. Yanagihara proved the following theorem at the ordinary meeting of Mathematical Institute in Tohoku University: Let E₀, E₁, E₂ be three congruent ovals which are situated in a homogetic positions and touch with each other. If we construct a ring of congruent ovals E₃, E₄, … around E₀, so that E_i, E_i-1, E₀ are located in homogetic positions and touch with each other, then E₆ touches E₁, that is, E₇ is the very oval E₁. If we denote the internal common tangent of E₀, E_i by t_i, then, by placing E₁ in a suitable position, we can make the three pairs of opposite sides of the convex hexagon formed by t₁, t₂, …, t₆ have respectively equal lengths. Cf. “Suni Zasso” (Miscellaneous notes in mathematics) 2, Tokyo-Butsuri-Gakko-Zasshi (Journal of Physics School) 26 (1917). The hexagon A₁A₂ … A₆ in Fig. 3 is applicable to the hexagon in the above mentioned Yanagihara's theorem, and the parallelogram in Fig. 4 is a special case of it.
(9) The determination of the arc to which Lemma 2 is applied, owes to Prof. Hombu. Our previous proof was as follows: If every side of P₁P₂ … P₆ is not of length D, then by applying the method of Lemma 2, or by repetitions of it to every side of base hexagon, if necessary, we can make two sides of the base hexagon be of length D. But in doing so the area of the asymmetric oval becomes smaller.
(10) We have considered the case when D and Δ or L and Δ are given, and not the case when D and L are so given that 3D≤L≤πD. For this case, by a property of the convex polygon in Favard's paper (Ann. Ecole Norm. 46, 1929), we get 2F≥D2{[π/2θ]sin2θ+sin(2θ[π/2θ])-√3} (≥DLcosθ-√3D2), where [ ] is Gauss's notation and θ is the root of [π/2θ]sinθ+cos(θ[π/2θ])=L/(2D) in the interval (0, π/6). The equality occurs when and only when the oval is an asymmetric curve whose base hexagon is a regular hexagon of sides D and whose breadth curve is a regular 6n-polygon inscribed in a circle with radius D. This inequality differs in some respects from Theorem 2 or 3, for the equality does not occur unless L=6nD sin (π/6n), n=1,2,3, ….
(*) Read at the annual meeting of the Math. Soc. of Japan held in June 2, 1951.
Added in proof by D. Hemmi. After we wrote this paper I received D. Ohmann's paper (Math. Z. 55, 1952. 347-352) and M. Sholander's (Trans. Amer. Math. Soc. 73, 1952. 139-173). The former does not touch the case 3D<L<πD and the latter gives the partial results and conjectures a property of the solution. The proofs of Sholander's conjecture and the inequality in Notes (10) may be seen in Bull. Yamagata Univ. (Natural science) 2 (1953) 157-170 and 3 (1953) 1-11.

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