  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".
