# Teorema Hobby-Rice 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 por 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 {displaystyle n+1} subintervals: {displaystyle 0=z_{0}Define a signed partition as a partition in which each subinterval {estilo de exibição eu} has an associated sign {displaystyle delta _{eu}} : {displaystyle delta _{1},dotsc ,delta _{k+1}in left{+1,-1certo}} The Hobby-Rice theorem says that for every n continuously integrable functions: {estilo de exibição g_{1},dotsc ,g_{n}colon [0,1]seta longa para a direita mathbb {R} } there exists a signed partition of [0,1] de tal modo que: {soma de estilo de exibição _{i=1}^{n+1}delta _{eu}!int_{z_{i-1}}^{z_{eu}}g_{j}(z),dz=0{texto{ por }}1leq jleq n.} (em outras palavras: 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 dentro 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". Anais da American Mathematical Society. Sociedade Americana de Matemática. 16 (4): 665–670. doi:10.2307/2033900. JSTOR 2033900. ^ Pinkus, Alan (1976). "A simple proof of the Hobby-Rice theorem". Anais da American Mathematical Society. Sociedade Americana de Matemática. 60 (1): 82-84. doi:10.2307/2041117. JSTOR 2041117. ^ Alon, Noga (1987). "Splitting Necklaces". Avanços em Matemática. 63 (3): 247-253. doi:10.1016/0001-8708(87)90055-7. ^ F.W. Simmons and F.E. Eles são (2003). "Consensus-halving via theorems of Borsuk-Ulam and Tucker" (PDF). Ciências Sociais Matemáticas. 45: 15-25. doi:10.1016/S0165-4896(02)00087-2. Este artigo sobre análise matemática é um esboço. Você pode ajudar a Wikipédia expandindo-a. Categorias: Theorems in measure theoryFair divisionCombinatorics on wordsTheorems in analysisMathematical analysis stubs

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