Kempf proved that when a point is unstable in the sense of geometric invariant theory, there is a "worst" destabilizing 1-parameter subgroup $\lambda$. It is natural to ask:

what are the worst destabilizing 1-ps for the unstable points in familiar GIT problems, such as those used to construct the moduli space of curves $M_g$? We first consider Chow points of rational curves with one unibranch singular point. Then the problem can be restated as an explicit problem in convex geometry (finding the proximum of a polyhedral cone to a point outside it). In computer experiments, we observe that combinatorial features of the worst 1-ps persist as the degree increases. This is joint work with Joshua Jackson (Sheffield).