Largest Inscribed Rectangles in Geometric Convex Sets

TL;DR

This paper develops optimization and geometric methods to find the largest inscribed rectangles and other shapes within convex sets, providing algorithms with improved efficiency and applicability in higher dimensions.

Contribution

It introduces new optimization models and algorithms for inscribed shape problems, including a unified framework and approximation algorithms with sub-linear running times.

Findings

Presented an interior-point method for higher-dimensional inscribed rectangles.

Developed a parametrized optimization approach for 2D convex sets.

Provided $(1- ext{epsilon})$-approximation algorithms with sub-linear complexity.

Abstract

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.