# 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 di A. 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 {stile di visualizzazione n+1} subintervals: {displaystyle 0=z_{0}Define a signed partition as a partition in which each subinterval {stile di visualizzazione i} has an associated sign {displaystyle delta _{io}} : {displaystyle delta _{1},punti ,delta _{k+1}in left{+1,-1Giusto}} The Hobby-Rice theorem says that for every n continuously integrable functions: {stile di visualizzazione g_{1},punti ,g_{n}colon [0,1]longrightarrow mathbb {R} } there exists a signed partition of [0,1] tale che: {somma dello stile di visualizzazione _{io=1}^{n+1}delta _{io}!int _{z_{i-1}}^{z_{io}}g_{j}(z),dz=0{testo{ per }}1leq jleq n.} (in altre parole: 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 in 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". Atti dell'American Mathematical Society. Società matematica americana. 16 (4): 665–670. doi:10.2307/2033900. JSTOR 2033900. ^ Pinkus, Allan (1976). "A simple proof of the Hobby-Rice theorem". Atti dell'American Mathematical Society. Società matematica americana. 60 (1): 82–84. doi:10.2307/2041117. JSTOR 2041117. ^ Alon, Noga (1987). "Splitting Necklaces". Progressi in matematica. 63 (3): 247–253. doi:10.1016/0001-8708(87)90055-7. ^ F.W. Simmons and F.E. Sono (2003). "Consensus-halving via theorems of Borsuk-Ulam and Tucker" (PDF). Scienze Matematiche Sociali. 45: 15–25. doi:10.1016/S0165-4896(02)00087-2. Questo articolo relativo all'analisi matematica è solo un abbozzo. Puoi aiutare Wikipedia espandendolo. Categorie: Theorems in measure theoryFair divisionCombinatorics on wordsTheorems in analysisMathematical analysis stubs

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