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".
| Original language | English (US) |
|---|---|
| Pages (from-to) | 52-81 |
| Number of pages | 30 |
| Journal | Journal of Approximation Theory |
| Volume | 156 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 2009 |
Bibliographical note
Funding Information:We thank Ofer Zeitouni for his comments on an earlier version of this manuscript, which led to revisiting the introduction as well as stating Remark 5.2 . We thank Immo Hahlomaa, Martin Mohlenkamp and the anonymous reviewers for the very careful reading of this manuscript and their constructive suggestions. We thank Paul Nevai for the professional handling of this manuscript. GL thanks Mark Green and IPAM (UCLA) for inviting him to take part in their program on multiscale geometry and analysis in high dimensions. This work has been supported by NSF grant #0612608.
Keywords
- Ahlfors regular measure
- Concentration inequalities
- Functional equations in several variables
- Geometric inequalities
- High-dimensional geometry
- Hypersine
- Polar sine
- Pre-Hilbert space
- Trigonometric identities
- d-semimetrics