Question: What is known about the problem of covering a finite set of $\mathbb{P}$ of points in the plane with convex polygons
- that have the same number $m$ of points from $\mathbb{P}$ as corners
- and exactly $k\gt 0$ points of $\mathbb{P}$ inside.
- corners may be shared among the convex polygons
- no two convex polygons may have the same set of inner points from $\mathbb{P}$
- only the points on the convex hull of $\mathbb{P}$ are not contined inside of the sought convex polygons
i.e. what are conditions for existence and algorithms for finding optimal sets of covering polygons, e.g. with minimal area sum?
The motivation for the question comes from an attempt to generalize $1D$ BSpline bases to scattered data; the analogy being that all BSpline basis functions for fixed degree splines have the same number of boundary and inner points as knots, also for higher dimensional tensor BSplines
