text: We consider the problem of finding inscribed boxes and axis-aligned inscribed boxes of maximum volume, inside a compact and solid convex set. 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 @math time, where @math is the dimension and @math is the number of inequalities defining the convex set. For the general case, after formulating the problem for higher dimensions we investigate the traditional 2-dimensional problem, which is in the literature merely considered for convex polygons, for a broad range of convex sets. We first present a new exact algorithm that finds the largest inscribed axis-aligned rectangle in such convex sets for any given direction of axes in @math time. Using this exact algorithm as a subroutine, we present an @math -approximation algorithm that computes @math -approximation to the largest inscribed rectangle with computational complexity of @math . Finally, we show that how this running time can be improved to @math with a @math pre-processing time when the convex set is a polygon.
  @cite_28: In the first part of this paper, we reduce two geometric optimization problems to convex programming: finding the largest axis-aligned box in the intersection of a family of convex sets, and finding the translation and scaling that minimizes the Hausdorff distance between two polytopes. These reductions imply that important cases of these problems can be solved in expected linear time. In the second part of the paper, we use convex programming to give a new, short proof of an interesting Helly-type theorem, first conjectured by Gru¨nbaum and Motzkin.
  @cite_22: We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O ( n 3 ) . We also give ( 1 - e ) -approximation algorithms that take time O ( e - 1 2 log ? n + e - 3 2 ) .
summary: To our knowledge, except Amenta's model @cite_28 for LIAR and the model of @cite_22 for LIR, no other optimization based algorithm published so far for these two problems. Again, except Amenta's model for LIAR @cite_28 , we are not aware of any model or algorithm for the higher dimension problem. Our optimization model for LIAR is much more efficient than Amenta's model. Also none of the algorithms in the literature are able to solve both problems, while our parametric approach is able to do so in 2-dimensional case. Moreover, except 's algorithm @cite_22 , no algorithm is published so far for either of these problems that is capable of dealing with other geometric convex sets such as ellipses and not just polygons. In 's algorithm @cite_22 it is not clear whether those two queries are doable for any convex set and if yes what would @math be in their algorithm's running time. Our algorithm is unique in the sense that its performance depends to the number of inequalities defining a convex set, e.g. while for a polygon this is @math for an ellipse this is one.