On Dihedral Angles of a Simplex

H. Maehara


For an $n$-simplex, let $\alpha,\,\beta$ denote the maximum, and the minimum dihedral angles of the simplex, respectively. It is proved that the inequality $\alpha\le \arccos(1/n)\le \beta$ always holds, and either side equality implies that the $n$-simplex is a regular simplex. Similar inequalities are also given for a star-simplex, which is defined as a simplex that has a vertex (apex) such that the angles between distinct edges incident to the apex are all equal. Further, an explicit formula for the dihedral angle of a star-simplex between two distinct facets sharing the apex in common is presented in terms of the angle between two edges incident to the apex.

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DOI: http://dx.doi.org/10.5539/jmr.v5n2p79

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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