  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.
