On approximation of asymmetric separators of the n-cube

Abstract

Abstract A new combinatorial result intertwined with the Brouwer fixed point theorem for the n-cube is given. This result can be used for any map (f 1, ..., fn): [0, 1] n → [0, 1] n to approximate the components of the set {(x 1, . . . , x n ) ∈ [0, 1] n : f i (x 1, . . . , x n ) = x i } that separate the n-cube between the i th opposite faces. Equivalently, for maps g : [0, 1] n → ℝ such that g(x)g(y) ≤ 0 for any x ∈ {0} × [0, 1] n-1and y ∈ {1} × [0, 1] n-1, one can use the algorithm to approximate the components of g -1(0) that separate [0, 1] n between {0} × [0, 1] n-1and {1} × [0, 1] n-1. The methods are based on an earlier result of P. Minc and the present authors and relate to results of several other authors such as Jayawant and Wong, Kulpa and Turzański, and Gale. Mathematics Subject Classification (2000): Primary 54H25; 54-04; Secondary 55M20; 54F55.