Largest Inscribed Rectangles in Geometric Convex Sets

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.

💡 Research Summary

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The paper addresses the problem of finding the maximum‑volume (or area) axis‑aligned rectangle that can be inscribed in a compact convex set defined by a finite collection of convex inequalities. The authors develop both geometric and optimization‑based approaches, provide new theoretical insights, and propose several algorithms with provable approximation guarantees.

Key contributions

1. Optimality properties of the MAIR in convex polygons – The authors prove that any maximum‑area inscribed rectangle (MAIR) in a convex polygon must fall into one of three configurations: (i) all four corners lie on the boundary, (ii) exactly one interior corner with an adjacent vertex‑corner, or (iii) the rectangle is a square with two opposite interior corners and two opposite vertex‑corners. The proof is simpler and stronger than previous results. Similar properties are extended to centrally symmetric and axially symmetric convex sets, marking the first such analysis for non‑polygonal convex bodies.

2. Optimization models for higher dimensions – By representing a box with its centre and half‑lengths along each axis, the authors formulate a convex program whose constraints are linear in the decision variables. This model avoids the exponential number of constraints that earlier formulations required. An interior‑point method is applied, and a full convergence and complexity analysis is provided. The resulting (1‑ε)‑approximation algorithm runs in
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