On two geometric inequalities


In inner product spaces the Ptolemaic inequality (1) and the quadrilateral inequality (2) are well known. By using the identity (3), we derive here (2) from (1). The rest of the paper is devoted to some comments on (1), (2), (3).

1. C. Alsina, J.L. Garcia-Roig, On a functional equation related to the Ptolemaic inequality, Aequationes Math. 84 (1987), 298-303.
2. M.M. Day, Normed Linear Spaces, Springer-Verlag, Berlin, 3rd edition, 1973.
3. H. Hornich, Eine Ungleichung für Vektorlängen, Math. Z. 48 (1942), 268-274.
4. L.M. Kelly, D.M. Smiley, M.F. Smiley, Two dimensional spaces and quadrilateral spaces, Amer. Math. Monthly 72 (1965), 753-754.
5. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality, Państwowe Wydawnictwo Naukowe, Warszawa, 1985.
6. S. Mazur, W. Orlicz, Grundlegende Eigenschaften der polynomischen Operationen, I, Studia Math. 5 (1934), 50-68.
7. D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin, 1970.
8. I.J. Schoenberg, A remark on M.M. Day's characterization of inner-product spaces and a conjecture of L.M. Blumenthal, Proc. Amer. Math. Soc. 3 (1953), 961-964.
9. D.M. Smiley, M.F. Smiley, The polygonal inequalities, Amer. Math. Monthly 71 (1964), 755-760.

Published : 1995-09-29

SimonA., & VolkmannP. (1995). On two geometric inequalities. Annales Mathematicae Silesianae, 9, 137-140. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14213

Alice Simon 
Départment de Mathématiques, Université d'Orléans, France  France
Peter Volkmann 
Mathematisches Institut I, Universität Karlsruhe, Germany  Germany

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