Before revision:  Using this exact algorithm as a subroutine, we present an \epsilon-approximation algorithm that computes (1-\epsilon)-approximation to the largest inscribed rectangle with computational complexity of O(\epsilon^{-1}n)}  . Finally, we show that how this running time can be improved to \mathcal{O(\epsilon^{-1}\log n) with a O(\epsilon^{-1}n) pre-processing time when the convex set is a polygon}  .
After revision:  Using this exact algorithm as a subroutine, we present an \epsilon-approximation algorithm that computes (1-\epsilon)-approximation to the largest inscribed rectangle with computational complexity of O(\epsilon^{-1}n)} . Finally, we show that how this running time can be improved to \mathcal{O(\epsilon^{-1}\log n) with a O(\epsilon^{-1}n) pre-processing time when the convex set is a polygon} a parametrized optimization approach is developed to find the largest (axis-aligned) inscribed rectangles in two-dimensional space .