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We prove that any minimal valid function for the k-dimensional innite group relaxation that is piecewise linear with at most k + 1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k = 1, and Cornu ejols and Molinaro for k = 2.

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