On dd-dimensional dd-semimetrics and simplex-type inequalities for high-dimensional sine functions

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