SUMS: Guo -- Lengths, Areas and Projections

This week’s SUMS talk is being given by Ivan Guo.  

Abstract: 
We first address the following seemingly obvious questions.  Suppose that one
box is placed inside another box.  Does the outside box necessarily have a larger
surface area? Does it have a larger sum of dimensions (length + width + height)? Surely
the answer is yes, but the reasons may not be so obvious.  

Next, we consider a cube in a random orientation.  What is the average size of the
shadow cast by the cube at noon? What is the average height of the cube? What if the
cube is replaced by another convex solid? Can we answer these questions in general? 

Remarkably, these two sets of geometric conundrums are closely connected via what is
known as Weyl’s Tube Formula.  An elementary version will be presented, which will
hopefully provide us with elegant solutions to the earlier questions.