# Selecting the Kth largest-area convex polygon

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## Abstract

The purpose of this note is to solve an open problem submitted by B. Chazelle [2]: Given a set *P* of *n* points in the Euclidean plane, select the *k*^{ th } largest-area convex polygon determined by subsets of *P*. We show that the decision problem is NP-hard by a reduction from the problem of finding the *k*^{ th } largest *m*-tuple [8] determined by *m* sets *X*_{1},*X*_{2},...,*X*_{ m }. We also show that the enumeration problem is #P-complete and exhibit a pseudo-polynomial-time algorithm for the decision problem.

## Keywords

Convex Hull Convex Polygon Euclidean Plane Good Subset Enumeration Problem
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## Copyright information

© Springer-Verlag Berlin Heidelberg 1989