Splitting necklaces

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Let N be an opened necklace with kaibeads of color i, 1 ≤ i ≤ t. We show that it is possible to cut N in (k - 1) · t places and partition the resulting intervals into k collections, each containing precisely aibeads of color i, 1 ≤ i ≤ t. This result is best possible and solves a problem of Goldberg and West. Its proof is topological and uses a generalization, due to Bárány, Shlosman and Szücs, of the Borsuk-Ulam theorem. By similar methods we obtain a generalization of a theorem of Hobby and Rice on L1-approximation. © 1987.




Alon, N. (1987). Splitting necklaces. Advances in Mathematics, 63(3), 247–253. https://doi.org/10.1016/0001-8708(87)90055-7

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