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We prove that any minimal valid function for the k-dimensional infinite grouprelaxation that is piecewise linear with at most k+1 slopes and does not factorthrough a linear map with non-trivial kernel is extreme. This generalizes atheorem of Gomory and Johnson for k=1, and Cornuejols and Molinaro for k=2.

A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation