The paper studies n-dimensional varieties in \({\mathbb{P}}^ N\) which are hypersurfaces in some linear subspace. These are called n-hypersurfaces. The first theorem in the paper states that if \(n>0\), then a subscheme of \({\mathbb{P}}^ N\) with the Hilbert polynomial of an n-hypersurface is indeed an n-hypersurface. It follows that such a Hilbert polynomial is minimal among all Hilbert polynomials of subschemes of the same dimension and degree. The second result is the following: Let X be a projective scheme with Hilbert polynomial \(P(t)=\sum^{n}_{i=0}a_ i\left( \begin{matrix} t+n-i\\ n-i\end{matrix} \right).\) Then \(2a_ 1\geq a_ 0(a_ 0-1)\), and if X is of pure dimension, equality implies that X is an n-hypersurface (of degree \(a_ 0)\).