Finding all Maximal Area Parallelograms in a Convex Polygon

We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an $n$-sided polygon in $O(n^2)$ time. Achieving this running time requires proving several new structural properties of the MAP. Our algorithm actually computes all the locally maximal area parallelograms (LMAPs). In addition to the algorithm, we prove that the LMAPs interleave each other, thus the number of LMAPs is bounded by $O(n)$. We discuss applications of our result to, among others, the problem of computing the maximum area centrally-symmetric convex body (MAC) inside a convex polygon, and the simplest case of the Heilbronn Triangle Problem.

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