On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions
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".
Cite
@article{arxiv.0805.1430,
title = {On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions},
author = {Gilad Lerman and Jonathan Tyler Whitehouse},
journal= {arXiv preprint arXiv:0805.1430},
year = {2009}
}
Comments
22 pages and 2 figures, updated to reflect publication in JAT and the DOI