Dan Klain (University of Massachusetts Lowell)
Volume Bounds for Shadow Covering
Suppose that K and L are compact convex subsets of n-dimensional
Euclidean space, and suppose that every (n-1)-dimensional orthogonal
projection (that is to say, every shadow) of L onto a subspace
contains a translate of the corresponding projection of K to that same
subspace.
This covering condition does not imply that L contains a translate of
K. In fact, we will see that it is even possible for L to have
strictly smaller volume! This leads to several questions:
- When does shadow covering imply that L contains a translate of K?
- What does shadow covering imply more generally about covering
relations between K and L?
- When does shadow covering imply that Vol(K) ≤ Vol(L)?
- What does shadow covering imply more generally about the ratio of
Vol(K) to Vol(L)?
The talk will address recent results concerning each of these
questions, as well as analogous results for projections onto subspaces
of some other fixed intermediate dimension. Open questions and
conjectures will also be posed.
Some of these results arise from joint work with Christina Chen and
Tanya Khovanova.
To the Combinatorics Seminar Web page.