Robert Connelly - Holes in a membrane: tension percolation

Consider a convex polygonal planar membrane made of inextendable material in three-space clamped on its boundary. Suppose a finite number of interior disjoint convex holes are created in the membrane. Depending on the configuration of holes, tension within the membrane can vanish on a region larger than just what is covered by the holes. We propose an algorithm that can calculate exactly those regions that must have no tension. This can be applied to the following problems.

1.  Consider a random process that creates holes in a given convex polygon in the plane. At what point does the unstressed portion occupy most of the region?

2.  Consider a finite collection of disjoint convex subsets of the plane. We find a tiling of the plane by convex sets, coming from the projection of a convex polyhedron in three-space, such that there is at least one, but a minimal number of subsets in each tile.

3.  Start with a triangular lattice grid in the plane. Remove each edge with probability p. What is the critical value of p, where the lattice fails to support a positive stress in any edge?

This is joint work with Joe Mitchell and Konstatin Rybnikov.