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In [[mathematics]], and in particular the [[necklace splitting problem]], the '''Hobby–Rice theorem''' is a result that is useful in establishing the existence of certain solutions.  It was proved in 1965 by Charles R. Hobby and [[John R. Rice (professor)|John R. Rice]];<ref name="Hobby 1965">{{cite journal| last=Hobby|first=C. R.|authorlink=John R. Rice (professor)|coauthors=Rice, J. R.|title=A moment problem in ''L''<sub>1</sub> approximation|journal=Proceedings of the American Mathematical Society|volume=16|pages=665–670| doi=10.2307/2033900| jstor=2033900| issue=4| publisher=American Mathematical Society| year=1965}}</ref> a simplified proof was given in 1976 by A. Pinkus.<ref name="Pinkus 1976">{{cite journal|last=Pinkus|first=Allan|title=A simple proof of the Hobby-Rice theorem|journal=Proceedings of the American Mathematical Society|volume=60|year=1976|pages=82–84|doi=10.2307/2041117|jstor=2041117|issue=1|publisher=American Mathematical Society}}</ref>


==The theorem==
If


: <math>g_1,\dotsc,g_k\colon[0,1]\longrightarrow\Bbb{R}</math>
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are given continuously integrable [[Function (mathematics)|functions]] then there exist
 
: <math>0=z_0 < z_1 < \dotsb < z_k < z_{k+1} = 1</math>
 
and
 
: <math>\delta_1,\dotsc,\delta_{k+1}\in\left\{+1,-1\right\}</math>
 
such that
 
: <math>\sum_{i=1}^{k+1}\delta_i\!\int_{z_{i-1}}^{z_i} g_j(z)\,dz=0\text{ for }1\leq j\leq k.</math>
 
The theorem was used by [[Noga Alon]] in the context of necklace splitting<ref name="Alon 1987">{{cite journal| last=Alon| first=Noga| title=Splitting Necklaces| journal=Advances in Mathematics| year=1987| volume=63| pages=247–253| doi=10.1016/0001-8708(87)90055-7| issue=3}}</ref> in 1987.
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Hobby-Rice theorem}}
[[Category:Measure theory]]
[[Category:Fair division]]
[[Category:Combinatorics on words]]
[[Category:Theorems in analysis]]
 
 
{{mathanalysis-stub}}

Latest revision as of 17:14, 27 December 2014


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