Applications to Euclidean Space

Abstract

We recall the definition of the

$$ standard\,n - sphere\quad {{\mathbb{S}}^n} = \{ x \in {{\mathbb{R}}^{{n + 1}}}|\left\| x \right\| = 1\} $$

and

$$ standard\,n - ball\quad {{\mathbb{B}}^n} = \{ y \in {{\mathbb{R}}^n}|\left\| y \right\| \leqslant 1\}, $$

where $||x|| = \sqrt {\sum\nolimits_{i = 0}^n {x_i^2} } .$. The open ball, $\mathop{{{{\mathbb{B}}^n}}}\limits^{ \circ } = \{ y \in {{\mathbb{R}}^n}|\left\| y \right\| < 1\}$ is also called standard n-cell. Let $Q = (0, \ldots, 0,1) \in {{\mathbb{S}}^n}$, the point with last coordinate Q _n = 1.