{"pk":65041,"title":"High-dimensional envy-free partitions","subtitle":null,"abstract":"A vast array of envy-free results have been found for the subdivision of one-dimensional resources, such as the interval \\([0,1]\\). The goal is to divide the space into \\(n\\) pieces and distribute them among \\(n\\) guests such that each receives their favorite pieces. We study high-dimensional versions of these results. We prove that several spaces of convex partitions of \\(\\mathbb{R}^d\\) allow for envy-free division among any \\(n\\) guests. We also prove the existence of convex partitions of \\(\\mathbb{R}^d\\) which allow envy-free divisions among several groups of \\(n\\) guests simultaneously.\n \nMathematics Subject Classifications: 91B32, 52A37, 55M20, 28A75\n \nKeywords: Mass partition, KKM cover, Envy-free partition, Equivariant topology, Voronoi diagram","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Mass partition"},{"word":"KKM cover"},{"word":"Envy-free partition"},{"word":"Equivariant topology"},{"word":"Voronoi diagram"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/4d6604q1","frozenauthors":[{"first_name":"Pablo","middle_name":"","last_name":"Soberón","name_suffix":"","institution":"Baruch College, City University of New York, New York, NY 10010, U.S.A. -- The Graduate Center, City University of New York, New York, NY 10016, U.S.A.","department":""},{"first_name":"Christina","middle_name":"","last_name":"Yu","name_suffix":"","institution":"Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.","department":""}],"date_submitted":"2026-04-20T19:38:19Z","date_accepted":"2026-04-20T19:38:19Z","date_published":"2026-04-20T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/65041/galley/49851/download/"}]}