# Thieves can make sandwiches

## TL;DR

This paper generalizes classical fair division theorems, proving the existence of equitable measure distributions among multiple parties using convex partitions, even when the number of measures exceeds the dimension.

## Contribution

It introduces a unified theorem extending the Ham Sandwich and Necklace Splitting theorems, with a novel topological and geometric approach for fair measure division.

## Key findings

- Existence of fair measure distributions among multiple thieves.
- Use of convex partitions roughly proportional to measure and dimension.
- Application of topological join and index theory in geometric partitioning.

## Abstract

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.