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.

翻译：本文考虑了在一个紧凑的锥形组中找到最大体积(轴对齐)的刻录框的问题,该组以数量有限的锥形不平等为定义,并展示了解决这些问题的优化和几何方法。开发了几种最优化模型,这些模型可以很容易地通用,以找到其他刻录的几何形状,如三角形、正方形和正方形。要找到在较高维度中最大轴对齐的刻录矩形,就提出并分析一个内部点法算法。对于二维空间,正在开发一种对称优化法,以找到在锥形组中刻录的最大矩形(轴对齐)的矩形组。优化法提供了一个解决广泛相关问题的统一框架。最后,提出了两种具有亚线性运行时间的计算几何 $( \ varepsilon)$- Apporomation 算法,从而改进了先前的结果。