# Hobby–Rice theorem Hobby–Rice theorem 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; a simplified proof was given in 1976 par un. Pinkus. The theorem Given an integer n, define a partition of the interval [0,1] as a sequence of numbers which divide the interval to {displaystyle n+1} subintervals: {displaystyle 0=z_{0}Define a signed partition as a partition in which each subinterval {style d'affichage i} has an associated sign {delta de style d'affichage _{je}} : {delta de style d'affichage _{1},pointsc ,delta _{k+1}in left{+1,-1droit}} The Hobby-Rice theorem says that for every n continuously integrable functions: {style d'affichage g_{1},pointsc ,g_{n}colon [0,1]longrightarrow mathbb {R} } there exists a signed partition of [0,1] tel que: {somme de style d'affichage _{je=1}^{n+1}delta _{je}!entier _{z_{i-1}}^{z_{je}}g_{j}(z),dz=0{texte{ pour }}1leq jleq n.} (autrement dit: for each of the n functions, its integral over the positive subintervals equals its integral over the negative subintervals). Application to fair division The theorem was used by Noga Alon in the context of necklace splitting dans 1987. Suppose the interval [0,1] is a cake. There are n partners and each of the n functions is a value-density function of one partner. We want to divide the cake into two parts such that all partners agree that the parts have the same value. This fair-division challenge is sometimes referred to as the consensus-halving problem. The Hobby-Rice theorem implies that this can be done with n cuts. References ^ Hobby, C. R; Rice, J. R. (1965). "A moment problem in L1 approximation". Actes de l'American Mathematical Society. Société mathématique américaine. 16 (4): 665–670. est ce que je:10.2307/2033900. JSTOR 2033900. ^ Pinkus, Alain (1976). "A simple proof of the Hobby-Rice theorem". Actes de l'American Mathematical Society. Société mathématique américaine. 60 (1): 82–84. est ce que je:10.2307/2041117. JSTOR 2041117. ^ Alon, Noga (1987). "Splitting Necklaces". Avancées en mathématiques. 63 (3): 247–253. est ce que je:10.1016/0001-8708(87)90055-7. ^ F.W. Simmons and F.E. Elles sont (2003). "Consensus-halving via theorems of Borsuk-Ulam and Tucker" (PDF). Sciences sociales mathématiques. 45: 15–25. est ce que je:10.1016/S0165-4896(02)00087-2. Cet article lié à l'analyse mathématique est un bout. Vous pouvez aider Wikipédia en l'agrandissant. Catégories: Theorems in measure theoryFair divisionCombinatorics on wordsTheorems in analysisMathematical analysis stubs

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