  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 $R^d$ 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 $R^d$ into convex parts, and
the computation of the Fadell-Husseini ideal valued index of the resulting
spaces.
