= Sphere in a normed vector space
{title2=$S_r(x_0)=\{x:\|x-x_0\|=r\}$}
In a <normed vector space>, this set consists of all points at a fixed positive <norm> distance $r$ from $x_0$. It also makes sense in an infinite-dimensional <Hilbert space>, unlike a specifically Euclidean <sphere>. In a real <Hilbert space>, the <sphere in a normed vector space> $\|u\|^2=2E_0>0$ has tangent variations $v$ satisfying $(u,v)=0$; this follows by differentiating the fixed squared <norm>. This is the geometric constraint used in <fixed-energy initial-condition optimality>.
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