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We show that the volume of the inner r-neighborhood of a polytope in the d-dimensional Euclidean space is a pluriphase Steiner-like function, i.e. a continuous piecewise polynomial function of degree d, thus proving a conjecture of Lapidus and Pearse. We discuss also the degree of differentiability of this function and give a lower bound in terms of the set of normal vectors of the hyperplanes defining the polytope. We also give sufficient conditions for the highest differentiability degree todoi:10.1090/s0002-9939-2011-11307-8 fatcat:e5v5xiwjpffwfgav7k366qa6ju