Let's say we have $n$ points say in $\mathbb{R}^2$, and $h$ points of the $n$ points where $n \ge h \ge 3$ are on the convex hull of this configuration.
How many ways are we able to create a non-convex set from this convex hull by moving at least one of the $h$ points on the convex hull? Would we need more information about the configuration? Is there a maximum number of the $h$ points that can be moved to create a non-convex set?