  For a set $S$ of $d$ points in the $n$-dimensional projective space over a
field of characteristic zero, we prove that the polynomials of degree $d$ whose
zero sets are cones over $S$ do not span the vector space of polynomials of
degree $d$ vanishing on $S$, if $d$ is odd and $d\ge 3$. Furthermore, they span
a subspace of codimension at least two, if $n=2$, $d=1\pmod 4$ and $d\ge 5$.
