トレミーの不等式
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2023/04/28 23:19 UTC 版)
ユークリッド幾何学において、トレミーの不等式（トレミーのふとうしき）とは、平面または高次元空間内の4点により決まる6つの距離についての不等式。4つの任意の点 A, B, C, D について、次の不等式が成り立つことを示す。
- ^ a b Schoenberg, I. J. (1940), “On metric arcs of vanishing Menger curvature”, Annals of Mathematics, Second Series 41: 715-726, doi:10.2307/1968849, MR0002903.
- ^ Steele, J. Michael (2004), “Exercise 4.6 (Ptolemy's Inequality)”, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, MAA problem books, Cambridge University Press, p. 69, ISBN 9780521546775.
- ^ Alsina, Claudi; Nelsen, Roger B. (2009), “6.1 Ptolemy's inequality”, When Less is More: Visualizing Basic Inequalities, Dolciani Mathematical Expositions, 36, Mathematical Association of America, pp. 82-83, ISBN 9780883853429.
- ^ Apostol (1967) attributes the inversion-based proof to textbooks by R. A. Johnson (1929) and Howard Eves (1963).
- ^ a b Stankova, Zvezdelina; Rike, Tom, eds. (2008), “Problem 7 (Ptolemy's Inequality)”, A Decade of the Berkeley Math Circle: The American Experience, MSRI Mathematical Circles Library, 1, American Mathematical Society, p. 18, ISBN 9780821846834.
- ^ a b Apostol, Tom M. (1967), “Ptolemy's inequality and the chordal metric”, Mathematics Magazine 40: 233-235, MR0225213.
- ^ Silvester, John R. (2001), “Proposition 9.10 (Ptolemy's theorem)”, Geometry: Ancient and Modern, Oxford University Press, p. 229, ISBN 9780198508250.
- ^ a b Giles, J. R. (2000), “Exercise 12”, Introduction to the Analysis of Normed Linear Spaces, Australian Mathematical Society lecture series, 13, Cambridge University Press, p. 47, ISBN 9780521653756.
- ^ Schoenberg, I. J. (1952), “A remark on M. M. Day's characterization of inner-product spaces and a conjecture of L. M. Blumenthal”, Proceedings of the American Mathematical Society 3: 961-964, doi:10.2307/2031742, MR0052035.
- ^ Howorka, Edward (1981), “A characterization of Ptolemaic graphs”, Journal of Graph Theory 5 (3): 323-331, doi:10.1002/jgt.3190050314, MR625074.
- ^ Buckley, S. M.; Falk, K.; Wraith, D. J. (2009), “Ptolemaic spaces and CAT(0)”, Glasgow Mathematical Journal 51 (2): 301-314, doi:10.1017/S0017089509004984, MR2500753.
[続きの解説]
「トレミーの不等式」の続きの解説一覧
- 1 トレミーの不等式とは
- 2 トレミーの不等式の概要
- 3 仮定と導出
- 4 4つの同一円上の点
- 5 一般距離空間
- トレミーの不等式のページへのリンク
