On two geometric inequalities



Abstract

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.
Download

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



The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.