Abstract 
In the first part of this paper, we reduce two geometric optimization problems to convex programming: finding the largest axisaligned 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 Hellytype theorem, first conjectured by Grunbaum and Motzkin.
