Lemma 3.4. $x_1,\dots ,x_d$ in ${ℝ}^n$ are not spherical if and only if there exists $c_i\in ℝ$, not all $0$, such that:

• (1) ${\sum }_i{c}_i=0$
• (2) ${\sum }_i{c}_i{x}_i=0$
• (3) ${\sum }_i{c}_i\parallel x_i{\parallel }_2^2\ne 0$