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arXiv · 0805.1430

On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions

Gilad Lerman·Jonathan Tyler Whitehouse
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Abstract

We show that high-dimensional analogues of the sine function (more precisely, the d-dimensional polar sine and the d-th root of the d-dimensional hypersine) satisfy a simplex-type inequality in a real pre-Hilbert space H. Adopting the language of Deza and Rosenberg, we say that these d-dimensional sine functions are d-semimetrics. We also establish geometric identities for both the d-dimensional polar sine and the d-dimensional hypersine. We then show that when d=1 the underlying functional equation of the corresponding identity characterizes a generalized sine function. Finally, we show that the d-dimensional polar sine satisfies a relaxed simplex inequality of two controlling terms "with high probability".

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Gilad Lerman, Jonathan Tyler Whitehouse. 2009-01-29. On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions. https://doi.org/10.1016/j.jat.2008.03.005

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