Wednesday, February 23rd, 2011
2:30pm - 4:00pm, in McCormack 2-116

Daniel Klain

UMass Lowell

If you can hide behind it, can you hide inside it?

Abstract: Suppose that K and L are compact convex subsets of Euclidean space, and suppose that, for every direction u, the orthogonal projection (that is, the shadow) of L onto the subspace orthogonal to u contains a translate of the corresponding projection of the body K. Does this imply that the original body L contains a translate of K? Can we even conclude that K has smaller volume than L? In other words, suppose K can ``hide behind'' L from any point of view (and without rotating). Does this imply that K can ``hide inside'' the body L? Or, if not, do we know, at least, that K has smaller volume? Although these questions have easily demonstrated negative answers in dimension 2 (since the projections are 1-dimensional, and convex 1-dimensional sets have very little structure), the (possibly surprising) answer to these questions continues to be No in Euclidean space of any finite dimension. In this talk I will give concrete constructions for convex sets K and L in n-dimensional Euclidean space such that each (n-1)-dimensional shadow of L contains a translate of the corresponding shadow of K, while at the same time K has strictly greater volume than L. This con- struction turns out to be sufficiently straightforward that a talented person could conceivably mold 3-dimensional examples out of modeling clay. The talk will then address a number of related questions, such as: under what additional conditions on K or L does shadow covering imply actual covering? What bounds can be placed on the volume ratio of K and L when the shadows of L cover those K?