Thieves can make sandwiches
Pavle V. M. Blagojević, Pablo Soberón · Bulletin of the London Mathematical Society · 2017
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 m r / 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.