A piecewise linear function can be described in different forms: as an arbitrarily nested expression of $\min$- and $\max$-functions, as a difference of two convex piecewise linear functions, or as a linear combination of maxima of affine-linear functions. In this paper, we provide two main results: first, we show that for every piecewise linear function there exists a linear combination of $\max$-functions with at most $n+1$ arguments, and give an algorithm for its computation. Moreover, these arguments are contained in the finite set of affine-linear functions that coincide with the given function in some open set. Second, we prove that the piecewise linear function $\max(0, x_{1}, \ldots, x_{n})$ cannot be represented as a linear combination of maxima of less than $n+1$ affine-linear arguments. This was conjectured by Wang and Sun in 2005 in a paper on representations of piecewise linear functions as linear combination of maxima.

翻译：分段线性函数可以用不同形式描述：作为$\min$和$\max$函数的任意嵌套表达式、作为两个凸分段线性函数的差、或者作为仿射线性函数最大值的线性组合。本文提供两个主要结果：首先，我们证明对于每个分段线性函数，存在一个至多包含$n+1$个参数的$\max$函数的线性组合，并给出其计算算法。此外，这些参数包含在与给定函数在某个开集上一致的有限仿射线性函数集合中。其次，我们证明分段线性函数$\max(0, x_{1}, \ldots, x_{n})$不能表示为少于$n+1$个仿射线性参数的最大值的线性组合。这一结论由Wang和Sun在2005年关于分段线性函数作为最大值线性组合表示的论文中提出猜想。