Convex equipartitions: the spicy chicken theorem |
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Authors: | Roman Karasev Alfredo Hubard Boris Aronov |
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Affiliation: | 1. Department of Mathematics, Moscow Institute of Physics and Technology, Institutskiy per. 9, 141700, Dolgoprudny, Russia 2. Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, 127994, Moscow, Russia 3. Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Sovetskaya st. 14, 150000, Yaroslavl, Russia 4. Département d’Informatique, école Normale Supérieure, 45 rue d’Ulm, 75005, Paris, France 5. Department of Computer Science and Engineering, Polytechnic Institute of NYU, Brooklyn, NY, 11201, USA
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Abstract: | We show that, for any prime power $n$ and any convex body $K$ (i.e., a compact convex set with interior) in $mathbb{R }^d$ , 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. |
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