A Combinatorial Topological Theorem and Its Application in Welfare Economics

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We prove the following combinatorial topological theorem : Let m and n be any positive integers with m*n and let T(n;1, 2)={x∈IR^n_+|Σ^n_<h=1>x_h=t, for some t with 1*t*2} be a subset of the n-dimensional Euclidean space IR^n. For every i=1, …, m, there is a class {C^j_i|j=1, …, n} of subsets C^j_i of T(n;1, 2). If C^j_i is closed for all i, j ; and if for every i=1, …, m, and every subset I of the set {1, 2, …, n}, the set F(I)={x∈T(n;1, 2)|Σ_<h∈I>x_h=Σ^n_<h=1>x_h} is a subset of the union of the sets C^j_i, j∈I, then there exists a connected subset C of the set T(n;1, 2) such that there exist x, y∈C with Σ^n_<h=1>x_h=1 and Σ^n_<h=1>y_h=2, and for every x∈C, there exists a partition II=(II(1), II(2), …, II(m)) of the set {1, 2, …, n} so that II(i)≠0 for every i and [numerical formula] This new result gives a substantial generalization of the classical lemma of Knaster, Kuratowski, and Mazurkiewicz (KKM Lemma) in combinatorial topology, and also leads to a generalization of Brouwer's fixed point theorem with a continuum of partition-based well-behaved inequalities. In addition, we present an application of this new result in welfare economics.
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