  A set $S\subset \mathbb{R}^n$ is a nonnegativity witness for a set $U$ of
real homogeneous polynomials if $F$ in $U$ is nonnegative on $\mathbb{R}^n$ if
and only if it is nonnegative at all points of $S$. We prove that the union of
the hyperplanes perpendicular to the elements of a root system $\Phi\subseteq
\mathbb{R}^n$ is a witness set for nonnegativity of forms of low degree which
are invariant under the reflection group defined by $\Phi$. We prove that our
bound for the degree is sharp for all reflection groups which contain
multiplication by $-1$. We then characterize subspaces of forms of arbitrarily
high degree where this union of hyperplanes is a nonnegativity witness set.
Finally we propose a conjectural generalization of Timofte's half-degree
principle for finite reflection groups.
