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The open mouth theorem in higher dimensions
- Source :
-
Linear Algebra & its Applications . Aug2012, Vol. 437 Issue 3, p1057-1069. 13p. - Publication Year :
- 2012
-
Abstract
- Abstract: Propositions 24 and 25 of Book I of Euclid’s Elements state the fairly obvious fact that if an angle in a triangle is increased without changing the lengths of its arms, then the length of the opposite side increases, and conversely. A satisfactory analogue that holds for orthocentric tetrahedra is established by S. Abu-Saymeh, M. Hajja, M. Hayajneh in a yet unpublished paper, where it is also shown that no reasonable analogue holds for general tetrahedra. In this paper, the result is shown to hold for orthocentric d-simplices for all . The ingredients of the proof consist in finding a suitable parametrization (by a single real number) of the family of orthocentric d-simplices whose edges emanating from a certain vertex have fixed lengths, and in making use of properties of certain polynomials and of Gram and positive definite matrices and their determinants. [Copyright &y& Elsevier]
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 437
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 75171215
- Full Text :
- https://doi.org/10.1016/j.laa.2012.03.012