  In this paper, we consider the homogeneous coordinate rings $A(Y_{n,d}) \cong
\mathbb{K}[\Omega_{n,d}]$ of monomial projections $Y_{n,d}$ of Veronese
varieties parameterized by subsets $\Omega_{n,d}$ of monomials of degree $d$ in
$n+1$ variables where: (1) $\Omega_{n,d}$ contains all monomials supported in
at most $s$ variables and, (2) $\Omega_{n,d}$ is a set of monomial invariants
of a finite diagonal abelian group $G \subset GL(n+1,\mathbb{K})$ of order $d$.
Our goal is to study when $\mathbb{K}[\Omega_{n,d}]$ is a quadratic algebra
and, if so, when $\mathbb{K}[\Omega_{n,d}]$ is Koszul or G-quadratic. For the
family (1), we prove that $\mathbb{K}[\Omega_{n,d}]$ is quadratic when $s \ge
\lceil \frac{n+2}{2} \rceil$. For the family (2), we completely characterize
when $\mathbb{K}[\Omega_{2,d}]$ is quadratic in terms of the group $G \subset
GL(3,\mathbb{K})$, and we prove that $\mathbb{K}[\Omega_{2,d}]$ is quadratic if
and only if it is Koszul. We also provide large families of examples where
$\mathbb{K}[\Omega_{n,d}]$ is G-quadratic.
