Before revision:  Our algorithms are capable of solving these two problems in any such set that can be represented with finite number of convex inequalities . For the axis-aligned case, we formulate the problem for higher dimensions and present an exact optimization algorithm which solves the problem in \mathcal{O(d^3+d^2n) time, where d is the dimension and n is the number of inequalities defining the convex set.
After revision:  Our algorithms are capable of solving these two problems in any such set that can be represented with finite number of convex inequalities , and presents an optimization approach for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and tetrahedrons. To find the largest axis-aligned case, we formulate the problem for higher dimensions and present an exact optimization algorithm which solves the problem in \mathcal{O(d^3+d^2n) time, where d is the dimension and n is the number of inequalities defining the convex set.