Minimizing a linear functional on a sphere
= Minimizing a linear functional on a sphere
{title2=$\min_{\|z\|=r}z\cdot w=-r\|w\|$}
For a nonzero Euclidean vector $w$, the <Cauchy-Schwarz inequality> gives $z\cdot w\geq-\|w\|$ when $\|z\|=1$. Equality is attained uniquely at $z=-w/\|w\|$. On a sphere of radius $r>0$ the minimizer is $-rw/\|w\|$, and the minimum is $-r\|w\|$. This turns many constrained vector problems into a normalization of one fixed vector. If $w=0$, every point minimizes the functional.