SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM
نویسندگان
چکیده
منابع مشابه
The Discrete Square Peg Problem
The square peg problem asks whether every Jordan curve in the plane has four points which form a square. The problem has been resolved (positively) for various classes of curves, but remains open in full generality. We present two new direct proofs for the case of piecewise linear curves.
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A continuous simple closed curve in the plane is also called a Jordan curve, and it is the same as an injective map from the unit circle into the plane or, equivalently, a topological embedding S1 ↩ R2. In its full generality Toeplitz’s problem is still open. So far it has been solved affirmatively for curves that are “smooth enough” by various authors for varying smoothness conditions; see the...
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The well-known “splitting necklace theorem” of Alon [1] says that each necklace with k · ai 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 g...
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15 صفحه اولSplitting Necklaces and Measurable Colorings of the Real Line
A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every k-colored necklace can be fairly split by at most k cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every k ≥ 1 there is a measura...
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ژورنال
عنوان ژورنال: Forum of Mathematics, Sigma
سال: 2020
ISSN: 2050-5094
DOI: 10.1017/fms.2019.51