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 pro...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2018-02, Vol.50 (1), p.108-123
Main Authors: Blagojević, Pavle V. M., Soberón, Pablo
Format: Article
Language:English
Subjects:
05A18 (secondary)
52A37
52A99
55S91 (primary)
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Summary: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.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms.12109