Suppose that the interiors of $n$ $m$-sided planar simple closed polygons generate a $\sigma$-algebra $A$.
How many atoms can $A$ possess, at the most?
Failing an exact answer, how about good bounds?
Suppose that the interiors of $n$ $m$-sided planar simple closed polygons generate a $\sigma$-algebra $A$.
How many atoms can $A$ possess, at the most?
Failing an exact answer, how about good bounds?
If I understood your question correctly, you want a bound on the primal shatter function of $m$-gons. For a little intro to the notion, see e.g., http://www.cs.cornell.edu/courses/cs4850/2009sp/Scribe%20Notes/Lecture%2034%20Monday%20April%2013.pdf
It is not hard to prove that the VC-dimension of convex $m$-gons is $2m+1$, so the answer to your question would be about $n^{2m+1}$.