Given a real algebraic group action on a linear space , a vector \(v\in V\) is called unstable if \(0\in \overline{Gv}-Gv\) , where the closure is taken with respect to the Zariski topology. A fundamental theorem of Kempf [21] in geometric invariant theory states that v is unstable if and only if there is a one-parameter subgroup A of G such that v is unstable with respect to it, i.e., \(0\in \overline{Av}-Av\) . Assuming G is a semisimple real algebraic group defined over \(\mathbb {Q}\) , we give a new proof to this result using a geometric interpretation of the setting. In the process, we also give a new proof of an effective version of this result by Shah and Yang [30, Prop. 2.2]. Our interpretation involves relating the length of vectors under a linear action to convex functions on certain \(\operatorname {CAT}(0)\) -spaces, and bound the latter from below by Busemann functions.
