# Maximum-Area Triangle in a Convex Polygon, Revisited

## TL;DR

This paper critically examines and improves algorithms for finding maximum-area inscribed polygons in convex polygons, revealing flaws in previous methods and proposing more accurate algorithms with different time complexities.

## Contribution

It demonstrates the failure of a 1979 linear-time algorithm for maximum-area inscribed triangles and introduces a corrected quadratic-time and an $O(n ext{log} n)$ divide-and-conquer algorithm.

## Key findings

- The 1979 linear-time algorithm for maximum-area inscribed triangles fails in some cases.
- A corrected quadratic-time algorithm is proposed for the problem.
- The 1979 algorithm for maximum-area $k$-gon with $k=4$ also fails to find the optimal solution.

## Abstract

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 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 $O(n\log n)$ time divide-and-conquer algorithm. Also we show by example that the algorithm presented in 1979 by Dobkin and Snyder for finding the largest-area $k$-gon that is inscribed in a convex polygon fails to find the optimal solution for $k=4$. Finally, we discuss the implications of our discoveries on the literature.