Splitting multidimensional necklaces

• Autores: Mark de Longueville, Rade T. Zivaljevic
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 218, Nº 3, 2008, págs. 926-939
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.02.003
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• Resumen

  • The well-known �splitting necklace theorem� of Alon [N. Alon, Splitting necklaces, Adv. Math. 63 (1987) 247�253] says that each necklace with kai beads of color i=1,�,n, can be fairly divided between k thieves by at most n(k-1) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [0,1] where beads of given color are interpreted as measurable sets Ai[0,1] (or more generally as continuous measures µi). We demonstrate that Alon's result is a special case of a multidimensional consensus division theorem about n continuous probability measures µ1,�,µn on a d-cube [0,1]d. The dissection is performed by m1++md=n(k-1) hyperplanes parallel to the sides of [0,1]d dividing the cube into m1md elementary cuboids (parallelepipeds) where the integers mi are prescribed in advance