The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity

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 1-PS for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric rational curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples.