On \(d\)-dimensional \(d\)-semimetrics and simplex-type inequalities for high-dimensional sine functions. (English) Zbl 1170.46025

Lerman, Gilad; Whitehouse, J. Tyler

J. Approx. Theory 156, No. 1, 52-81 (2009).

Summary: 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 M.–M. Deza and I. G. Rosenberg [Eur. J. Comb. 21, No. 6, 797–806 (2000; Zbl 0988.54029)], 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”.

MSC:

46C99	Inner product spaces and their generalizations, Hilbert spaces
52C99	Discrete geometry
39B05	General theory of functional equations and inequalities
54E25	Semimetric spaces
51M16	Inequalities and extremum problems in real or complex geometry

Keywords:

high-dimensional geometry; polar sine; hypersine; \(d\)-semimetrics; geometric inequalities; concentration inequalities; pre-Hilbert space; functional equations in several variables; trigonometric identities; Ahlfors regular measure

Citations:

Zbl 0988.54029

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