Largest triangles in a polygon

TL;DR

This paper explores various versions of the problem of finding maximum-area inscribed triangles in polygons, providing exact algorithms and approximation methods for different constraints and polygon types.

Contribution

It introduces exact algorithms for eight versions of the maximum inscribed triangle problem and approximation algorithms for convex polygons with reorientations.

Findings

Exact algorithms for all problem variants.

Approximation algorithms for convex polygons with reorientations.

Comprehensive analysis of triangle inscribed in polygons.

Abstract

We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane. We consider eight versions of the problem: we use either convex polygons or simple polygons as the container; we require the triangles to have either one corner with a fixed angle or all three corners with fixed angles; we either allow reorienting the triangle or require its orientation to be fixed. We present exact algorithms for all versions of the problem. In the case with reorientations for convex polygons with $n$ vertices, we also present $(1-\varepsilon)$-approximation algorithms.