**Questions:**

Given a set $\mathcal{P}$ of points in the Euclidean plane, what is the complexity of finding for a given point $p_i$

*inside*the convex hull $\mathrm{CH}\left(\mathcal{P}\right)$ the set $\lbrace q_{ij}\rbrace\subseteq\mathcal{P}\setminus p_i\ $ that minimizes $\sum\limits_{j}\left\|q_{ij}-p_i\right\|$ and for which every line through $p_i$ has points from $\lbrace q_{ij} \rbrace$ to both sides?is the subgraph induced by the edge-set set $\lbrace\lbrace p_i, q_{ij}\rbrace\rbrace$ connected?

remark:

$\bigcup\limits_i\bigcup\limits_j (p_i,q_{ij})$ partitions the convex-combination of $\mathcal{P}$ into convex polygonal regions whose minimum-weight triangulation together with the edges of the convex hull $\mathrm{CH}(\mathcal{P})$ and $\bigcup\limits_i\bigcup\limits_j (p_i,q_{ij})$ may yield a good approximation of $\mathcal{P}$'s minimum-weight triangulation.