Finding C-convex holes in a planar point set
In "Counting Convex Polygons in Planar Point Sets" (link) Mitchell et al. show an algorithm which can do this in time $O(cn^3)$.

Note however that if you need to report those polygons then you can't do better than $O(n^c)$ as there might be $\Omega(n^c)$ such polygons to begin with (consider for example $n$ points in convex position).