# Largest Inscribed Rectangles in Geometric Convex Sets

arXiv: https://arxiv.org/abs/1905.13246 · HTML: https://mlsanity.com/paper/1905.13246/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 May 2019.

## Built on by (most cited first, 1 of 1)

- [Certifying Robustness of Learning-Based Keypoint Detection and Pose Estimation Methods](https://mlsanity.com/paper/2408.00117/) (2024)

## Abstract (arXiv)

This paper considers the problem of finding maximum volume (axis-aligned) inscribed boxes in a compact convex set, defined by a finite number of convex inequalities, and presents optimization and geometric approaches for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and squares. To find the largest axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed. For 2-dimensional space, a parametrized optimization approach is developed to find the largest (axis-aligned) inscribed rectangles in convex sets. The optimization approach provides a uniform framework for solving a wide variety of relevant problems. Finally, two computational geometric (1-varepsilon)--approximation algorithms with sub-linear running times are presented that improve the previous results.