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

TL;DR

This paper demonstrates that high-dimensional sine functions, specifically the d-dimensional polar sine and hypersine, satisfy simplex-type inequalities in real pre-Hilbert spaces, extending classical sine properties to higher dimensions.

Contribution

It introduces the concept of d-semimetrics as high-dimensional sine functions satisfying simplex inequalities and establishes geometric identities and probabilistic inequalities for these functions.

Findings

High-dimensional sine functions satisfy simplex-type inequalities.

Geometric identities are established for d-dimensional polar sine and hypersine.

The d-dimensional polar sine satisfies a relaxed inequality with high probability.

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".