Skip to main content

On Simple Polygonalizations with Optimal Area


We discuss the problem of finding a simple polygonalization for a given set of vertices P that has optimal area. We 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 and 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. This answers a generalization of a question stated by Suri in 1989. Finally, we turn to higher dimensions, where we prove that, for 1 \(\leq\) k d , 2 \(\leq\) d , it is NP-hard to determine the smallest possible total volume of the k -dimensional faces of a d -dimensional simple nondegenerate polyhedron with a given vertex set, answering a generalization of a question stated by O'Rourke in 1980.

Author information

Authors and Affiliations


Additional information

Received June 26, 1997, and in revised form February 13, 1999, and May 19, 1999.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fekete, S. On Simple Polygonalizations with Optimal Area . Discrete Comput Geom 23, 73–110 (2000).

Download citation

  • Issue Date:

  • DOI:


  • High Dimension
  • Maximum Weight
  • Minimum Weight
  • Simple Polygon
  • Optimal Area