Maximum-area triangle in a convex polygon, revisited

Maximum-Area Triangle in a Convex Polygon, Revisited

We revisit the following problem: Given a convex polygon P , find the largest-area inscribed triangle. We show by example that the linear-time algorithm presented in 1979 by Dobkin and Snyder [1] for solving this problem fails. We then proceed to show that with a small adaptation, their approach does lead to a quadratic-time algorithm. We also present a more involved O(n logn) time divide-and-c...

Maximum-Area Quadrilateral in a Convex Polygon, Revisited

In this note we show by example that the algorithm presented in 1979 by Dobkin and Snyder [1] for finding the largest-area k-gon that is inscribed in a convex polygon fails to find the optimal solution for k = 4. This question, posed by Keikha et al. [2] who show that the presented method fails to find the optimal solution for k = 3.

Finding the Maximum Area Centrosymmetric Polygon in a Convex Polygon

A linear-time algorithm for the maximum-area inscribed triangle in a convex polygon

Given the n vertices of a convex polygon in cyclic order, can the triangle of maximum area inscribed in P be determined by an algorithm with O(n) time complexity? A purported linear-time algorithm by Dobkin and Snyder from 1979 has recently been shown to be incorrect by Keikha, Löffler, Urhausen, and van der Hoog. These authors give an alternative algorithm with O(n log n) time complexity. Here...

Finding the Maximum Area Parallelogram 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(n2) time. Achieving this running time requires proving several new structural properties of the MAP, and combining them with a rotating technique of Toussaint [10]. We also discuss applications of our result to th...