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 inside a convex polygon, and the simplest case of the Heilbronn Triangle Problem .
Polygon inclusion problems have been studied extensively in geometric optimization. In this paper, we consider the variant of finding the maximum area parallelograms (MAPs) and all the locally maximal area parallelograms (LMAPs) inside a given convex polygon. Our main result is an O(n^2) time algorithm, where n denotes the number of edges of the given polygon . Achieving this running time requires proving several structural properties of the LMAPs. As an additional result , we prove that the LMAPs interleave each other, thus we bound the number of LMAPs to be O(n). We discuss applications of our result to, among others, the problem of computing the maximum area centrally-symmetric convex body inside a convex polygon, and the simplest case of the Heilbronn triangle problem .
R
We consider the problem
Polygon inclusion problems have been studied extensively in geometric optimization. In this paper, we consider the variant
R
parallelogram (MAP)
parallelograms (MAPs) and all the locally maximal area parallelograms (LMAPs)
D
algorithm for computing the MAP in an n-sided polygon in
A
algorithm, where n denotes the number of edges of the given polygon
D
new
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MAP. Our algorithm actually computes all the locally maximal area parallelograms (LMAPs). In addition to the algorithm
LMAPs. As an additional result
A
we bound
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is bounded by
to be
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Triangle Problem
triangle problem