Separation properties for closures of toric orbits

A subset $X$ of a vector space $V$ is said to have the ‘separation property’ if it separates linear forms in the following sense: for each pair $(\alpha,\beta)$ of linearly independent forms on $V$ there exists a point $x\in X$ such that $\alpha(x)=0$ and $\beta(x)\ne0$; equivalently, each homogeneous hyperplane $H\subseteq V$ is linearly spanned by its intersection with $X$.
For orbit closures in representation spaces of an algebraic torus a criterion for the separation property is obtained. Strong and weak separation properties are also considered.
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