Thieves can make sandwiches

We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of m measures in Rd among r thieves using roughly mr/d convex pieces, even in the cases when m is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of Rd into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces.