= Gale hemisphere lemma
{c}
For integers $n=2k+d$ with $k\geq1$ and $d\geq0$, there are $n$ labelled points of $S^d$ such that every <open hemisphere> contains at least $k$ of them. The normalized vectors $(-1)^i(1,t_i,\ldots,t_i^d)$ for increasing $t_i$ give a signed <moment curve> construction. A nonzero <polynomial> of <polynomial degree> at most $d$ cannot have enough sign changes for fewer than $k$ positive alternating sample values. Zero sample values can be perturbed negatively by a <Lagrange interpolation polynomial>. Labels allow repeated positions when $d=0$.
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