Affine hyperplane ($H=\{v\in V:\ell (v)=c\},\ell \ne 0$)

In a finite-dimensional real vector space, a level set of a nonzero linear functional is an affine hyperplane. Choosing $v_0$ in that level set gives $H=v_0+\ker \ell$, so its dimension is one less than that of the ambient space. With a Euclidean inner product, translation followed by an orthonormal basis of $\ker \ell$ identifies it isometrically with the lower-dimensional Euclidean space. The coordinate-sum constraint on balanced cut vectors is an example.