Convex Equipartitions: The Spicy Chicken Theorem

• Published in 2013
• Added on 2022-04-24

We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into n convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov-Borsuk-Ulam theorem for convex sets in the model spaces of constant curvature. Most of the results in this paper appear in arxiv:1011.4762 and in arxiv:1010.4611. Since the main results and techniques there are essentially the same, we have merged the papers for journal publication. In this version we also provide a technical alternative to a part of the proof of the main topological result that avoids the use of compactly supported homology.

Links

• http://arxiv.org/abs/1306.2741v2
• http://arxiv.org/pdf/1306.2741v2

Other information

BibTeX entry

@article{ConvexEquipartitionsTheSpicyChickenTheorem,
	key = {ConvexEquipartitionsTheSpicyChickenTheorem},
	type = {article},
	title = {Convex Equipartitions: The Spicy Chicken Theorem},
	author = {Roman Karasev and Alfredo Hubard and Boris Aronov},
	abstract = {We show that, for any prime power n and any convex body K (i.e., a compact
convex set with interior) in Rd, there exists a partition of K into n convex
sets with equal volumes and equal surface areas. Similar results regarding
equipartitions with respect to continuous functionals and absolutely continuous
measures on convex bodies are also proven. These include a generalization of
the ham-sandwich theorem to arbitrary number of convex pieces confirming a
conjecture of Kaneko and Kano, a similar generalization of perfect partitions
of a cake and its icing, and a generalization of the Gromov-Borsuk-Ulam theorem
for convex sets in the model spaces of constant curvature.
  Most of the results in this paper appear in arxiv:1011.4762 and in
arxiv:1010.4611. Since the main results and techniques there are essentially
the same, we have merged the papers for journal publication. In this version we
also provide a technical alternative to a part of the proof of the main
topological result that avoids the use of compactly supported homology.},
	comment = {},
	date_added = {2022-04-24},
	date_published = {2013-12-07},
	urls = {http://arxiv.org/abs/1306.2741v2,http://arxiv.org/pdf/1306.2741v2},
	collections = {food,fun-maths-facts},
	url = {http://arxiv.org/abs/1306.2741v2 http://arxiv.org/pdf/1306.2741v2},
	year = 2013,
	urldate = {2022-04-24},
	archivePrefix = {arXiv},
	eprint = {1306.2741},
	primaryClass = {math.MG}
}