# Extreme functions with an arbitrary number of slopes

## TL;DR

This paper constructs a sequence of extreme valid functions with an arbitrary number of slopes, demonstrating that extreme functions can be highly complex and countering previous conjectures about their structure.

## Contribution

It introduces a method to generate extreme functions with any number of slopes, including a continuous limit with infinitely many slopes, advancing understanding of their complexity.

## Key findings

- Constructed extreme functions with any number of slopes
- Provided a continuous extreme function with infinitely many slopes
- Countered the conjecture that all extreme functions are piecewise linear

## Abstract

For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number $k\geq 2$, there is a function in the sequence with $k$ slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson stating that all extreme functions are piecewise linear. These constructions are extended to obtain functions for the higher dimensional group problems via the sequential-merge operation of Dey and Richard.