Area optimization of simple polygons

We discuss problems of optimizing the area of a simple polygon for a given set of vertices P and show that these problems are very closely related to problems of optimizing the number of points from a set Q in a simple polygon with vertex set P. We prove that it is NP-complete to find a minimum weight polygon or a maximum weight polygon for a given vertex set, resulting in a proof of NP-completeness for the corresponding area optimization problems. We show that we can find a polygon of more than half the area <italic>AR(conv(P))</italic> of the convex hull <italic>conv(P)</italic> of P, and demonstrate that it is NP-complete to decide whether there is a simple polygon of at least (3/2 + ε)<italic>AR(conv(P))</italic>. Finally, we prove that for 1 ≤ <italic>k</italic> ≤ <italic>d</italic>, 2 ≤ <italic>d</italic>, it is NP-hard to minimize the volume of the <italic>k</italic>-dimensional faces of a <italic>d</italic>-dimensional simple non-degenerate polyhedron with a given vertex set, answering a generalization of a question stated by O'Rourke in 1980.

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