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71:
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31:
is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by
Charles R. Hobby and
878:
762:
830:
24:
603:
partners agree that the parts have the same value. This fair-division challenge is sometimes referred to as the
628:
558:{\displaystyle \sum _{i=1}^{n+1}\delta _{i}\!\int _{z_{i-1}}^{z_{i}}g_{j}(z)\,dz=0{\text{ for }}1\leq j\leq n.}
32:
720:
572:
functions, its integral over the positive subintervals equals its integral over the negative subintervals).
868:
599:
functions is a value-density function of one partner. We want to divide the cake into two parts such that
810:
696:
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814:
779:
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729:
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205:
76:
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715:
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188:{\displaystyle 0=z_{0}<\underbrace {z_{1}<\dotsb <z_{n}} <z_{n+1}=1}
20:
750:
581:
802:
325:{\displaystyle \delta _{1},\dotsc ,\delta _{k+1}\in \left\{+1,-1\right\}}
783:
700:
659:
407:{\displaystyle g_{1},\dotsc ,g_{n}\colon \longrightarrow \mathbb {R} }
691:
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problem. The Hobby–Rice theorem implies that this can be done with
755:"Consensus-halving via theorems of Borsuk-Ulam and Tucker"
818:
426:
348:
258:
228:
208:
102:
79:
53:
35:; a simplified proof was given in 1976 by A. Pinkus.
47:
of the interval as a division of the interval into
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406:
324:
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187:
85:
65:
464:
679:Proceedings of the American Mathematical Society
640:Proceedings of the American Mathematical Society
417:there exists a signed partition of such that:
584:in the context of necklace splitting in 1987.
838:
646:(4). American Mathematical Society: 665–670.
73:subintervals by as an increasing sequence of
8:
685:(1). American Mathematical Society: 82–84.
631:; Rice, J. R. (1965). "A moment problem in
335:The Hobby–Rice theorem says that for every
845:
831:
675:"A simple proof of the Hobby–Rice theorem"
733:
690:
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202:as a partition in which each subinterval
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78:
52:
620:
7:
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339:continuously integrable functions:
817:. You can help Knowledge (XXG) by
14:
568:(in other words: for each of the
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510:
396:
393:
381:
1:
776:10.1016/S0165-4896(02)00087-2
763:Mathematical Social Sciences
735:10.1016/0001-8708(87)90055-7
576:Application to fair division
884:Mathematical analysis stubs
587:Suppose the interval is a
242:{\displaystyle \delta _{i}}
900:
864:Theorems in measure theory
796:
25:necklace splitting problem
16:Necklace splitting problem
595:partners and each of the
580:The theorem was used by
23:, and in particular the
721:Advances in Mathematics
222:has an associated sign
874:Combinatorics on words
813:–related article is a
673:Pinkus, Allan (1976).
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811:mathematical analysis
716:"Splitting Necklaces"
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879:Theorems in analysis
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714:Alon, Noga (1987).
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66:{\displaystyle n+1}
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29:Hobby–Rice theorem
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749:F.W. Simmons and
605:consensus-halving
535:
215:{\displaystyle i}
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86:{\displaystyle n}
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200:signed partition
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728:(3): 247–253.
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869:Fair division
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819:expanding it
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629:Hobby, C. R.
623:
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591:. There are
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33:John R. Rice
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784:10419/94656
39:The theorem
21:mathematics
858:Categories
615:References
770:: 15–25.
582:Noga Alon
547:≤
541:≤
479:−
467:∫
456:δ
429:∑
397:⟶
379::
363:…
312:−
295:∈
280:δ
273:…
261:δ
231:δ
198:Define a
156:⏟
139:⋯
93:numbers:
45:partition
43:Define a
753:(2003).
751:F.E. Su
701:2041117
660:2033900
699:
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