Problem: Let $Q$ be a convex quadrilateral which is cut into convex pieces (cells) by a finite number of lines. For any collection $(Q_i)_1^n$ of these cells, decompose $Q$ into nonoverlapping convex polygons $(R_i)_1^n$ so that $Q_i \subset R_i$ for every $i$, and $\sum s_i \le 4n$, where $s_i$ denotes the number of sides of $R_i$.
This is a technical lemma I need for a separate problem, but I can really make progress on it.
Note the following:
Dissecting a convex polygon with a straight line produces two convex polygons.
For any two cells in $Q$, at least one of the original cutting lines separates the two cells from each other.
Using these two observations, we can recursively subdivide $Q$ using the following procedure:
Choose any two of the $Q_i$, then choose any cutting line which separates them. This line separates $Q$ into two convex regions. If any of these two regions contains more than one of the $Q_i$ then we again choose two of the $Q_i$ which are contained within the offending region, choose a cutting line separating them and subdivide this region with the chosen line. Continue until all regions contain only one of the $Q_i$.
Fedja's comment explains why $\sum s_i\le 4n$ for the resulting decomposition.