 
Summary: Bounded boxes, Hausdorff distance, and a new proof of an interesting
Hellytype theorem.
Nina Amenta \Lambda
The Geometry Center
1300 South Second Street
Minneapolis, MN. 55454
Abstract
In the first part of this paper, we reduce two geometric
optimization problems to convex programming: find
ing 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 interest
ing Hellytype theorem, first conjectured by Gr¨unbaum
and Motzkin.
1 Introduction
Linear programming is a popular tool in computa
