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 u? normal 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 construction 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 if the shadows of L cover those K?