# Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the *k*-dimensional case with applications to \(k=2\)

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## Abstract

We develop foundational tools for classifying the extreme valid functions for the *k*-dimensional infinite group problem. In particular, we present the general regular solution to Cauchy’s additive functional equation on restricted lower-dimensional convex domains. This provides a *k*-dimensional generalization of the so-called Interval Lemma, allowing us to deduce affine properties of the function from certain additivity relations. Next, we study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. We then give a theory of non-extremality certificates in the form of perturbation functions. We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function.

## Mathematics Subject Classification

90C10 90C57 39B52 39B62## References

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